Chapter XIV.


Music was forced first to select artistically, and then to shape for itself, the material on which it works. Painting and sculpture find the fundamental character of their materials, form and colour, in nature itself, which they strive to imitate. Poetry finds its materials ready formed in the words of language. Architecture has, indeed, also to create its own forms; but they are partly forced upon it by technical and not by purely artistic considerations. Music alone finds an infinitely rich but totally shapeless plastic material in the tones of the human voice and artificial musical instruments, which must be shaped on purely artistic principles, unfettered by any reference to utility as in architecture, or to the imitation of nature as in the fine arts, or to the existing symbolical meaning of sounds as in poetry. There is a greater and more absolute freedom in the use of the material for music than for any other of the arts. But certainly it is more difficult to make a proper use of absolute freedom, than to advance where external irremovable landmarks limit the width of the path which the artist has to traverse. Hence also the cultivation of the tonal material of music has, as we have seen, proceeded much more slowly than the development of the other arts.

It is now our business to investigate this cultivation.

The first fact that we meet with in the music of all nations, so far as is yet known, is that alterations of pitch in melodies take place by intervals, and not by continuous transitions. The psychological reason of this fact would seem to be the same as that which led to rhythmic subdivision periodically repeated. All melodies are motions within extremes of pitch. The incorporeal material of tones is much more adapted for following the musician's intention in the most delicate and pliant manner for every species of motion, than any corporeal material, however light. Graceful rapidity, grave procession, quiet advance, wild leaping, all these different characters of motion and a thousand others in the most varied combinations and degrees, can be represented by successions of tones. And as music expresses these motions, it gives an expression also to those mental conditions which naturally evoke similar motions, whether of the body and the voice, or of the thinking and feeling principle itself. Every motion is an expression of the power which produces it, and we instinctively measure the motive force by the amount of motion which it produces. This holds equally and perhaps more for the motions due to the exertion of power by the human will and human impulses, than for the mechanical motions of external nature. In this way melodic progression can become the expression of the most diverse conditions of human disposition, not precisely of human feelings[1] but at least of that state of sensitiveness which is produced by feelings. In English the phrase out of tune, unstrung, and in German the word stimmung, literally tuning, are transferred from music to mental states. The words are meant to denote those peculiarities of mental condition which are capable of musical representation. I think we might appropriately define gemüthsstimmung, or mental tune, as representing that general character temporarily shewn by the motion of our conceptions, and correspondingly impressed on the motions of our body and voice. Our thoughts may move fast or slowly, may wander about restlessly and aimlessly in anxious excitement, or may keep a determinate aim distinctly and energetically in view; they may lounge about without care or effort in pleasant fancies, or, driven back by some sad memories, may return slowly and heavily from the spot with short weak steps. All this may be imitated and expressed by the melodic motion of the tones, and the listener may thus receive a more perfect and impressive image of the 'tune' of another person's mind, than by any other means, except perhaps by a very perfect dramatic representation of the way in which such a person really spoke and acted.

Aristotle also formed a similar conception of the effect of music. In his 29th problem he says: 'Why do rhythms and melodies, which are composed of sound, resemble the feelings; while this is not the case for tastes, colours, or smells? Can it be because they are motions, as actions are also motions? Energy itself belongs to feeling and creates feeling. But tastes and colours do not act in the same way.' And at the end of the 27th problem he says: 'These motions, i.e. rhythms and melodies, are active, and action is the sign of feeling.' [2]

Not merely music but even other kinds of motions may produce similar effects. Water in motion, as in cascades or sea waves, has an effect in some respects similar to music. How long and how often can we sit and look at the waves rolling in to shore! Their rhythmic motion, perpetually varied in detail, produces a peculiar feeling of pleasant repose or weariness, and the impression of a mighty orderly life, finely linked together. When the sea is quiet and smooth we can enjoy its colouring for a while, but this gives no such lasting pleasure as the rolling waves. Small undulations, on the other hand, on small surfaces of water, follow one another too rapidly, and disturb rather than please.

But the motion of tone surpasses all motion of corporeal masses in the delicacy and ease with which it can receive and imitate the most varied descriptions of expression. Hence it arrogates to itself by right the representation of states of mind, which the other arts can only indirectly touch by shewing the situations which caused the emotion, or by giving the resulting words, acts, or outward appearance of the body. The union of music to words is most important, because words can represent the cause of the frame of mind, the object to which it refers, and the feeling which lies at its root, while music expresses the kind of mental transition which is due to the feeling. When different hearers endeavour to describe the impression of instrumental music, they often adduce entirely different situations or feelings which they suppose to have been symbolised by the music. One who knows nothing of the matter is then very apt to ridicule such enthusiasts, and yet they may have been all more or less right, because music does not represent feelings and situations, but only frames of mind, which the hearer is unable to describe except by adducing such outward circumstances as he has himself noticed when experiencing the corresponding mental states. Now different feelings may occur under different circumstances and produce the same states of mind in different individuals, while the same feelings may give rise to different states of mind. Love is a feeling. But music cannot represent it directly as such. The mental states of a lover may, as we know, shew the extremest variety of change. Now music may perhaps express the dreamy longing for transcendent bliss which love may excite. But precisely the same state of mind might arise from religious enthusiasm. Hence when a piece of music expresses this mental state it is not a contradiction for one hearer to find in it the longing of love, and another the longing of enthusiastic piety. In this sense Vischer's rather paradoxical statement that the mechanics of mental emotion are perhaps best studied in their musical expression, may be not altogether incorrect. We really possess no other means of expressing them so exactly and delicately.

As we have seen, then, melody has to express a motion, in such a manner that the hearer may easily, clearly, and certainly appreciate the character of that motion by immediate perception. This is only possible when the steps of this motion, their rapidity and their amount, are also exactly measurable by immediate sensible perception. Melodic motion is change of pitch in time. To measure it perfectly, the length of time elapsed, and the distance between the pitches, must be measurable. This is possible for immediate audition only on condition that the alterations both in time and pitch should proceed by regular and determinate degrees. This is immediately clear for time, for even the scientific, as well as all other measurement of time, depends on the rhythmical recurrence of similar events, the revolution of the earth or moon, or the swings of a pendulum. Thus also the regular alternation of accentuated and unaccentuated sounds in music and poetry gives the measure of time for the composition. But whereas in poetry the construction of the verse serves only to reduce the external accidents of linguistic expression to artistic order; in music, rhythm, as the measure of time, belongs to the inmost nature of expression. Hence also a much more delicate and elaborate development of rhythm was required in music than in verse.

It was also necessary that the alteration of pitch should proceed by intervals, because motion is not measurable by immediate perception unless the amount of space to be measured is divided off into degrees. Even in scientific investigations we are unable to measure the velocity of continuous motion except by comparing the space described with the standard measure, as we compare time with the seconds pendulum.

It may be objected that architecture in its arabesques, which have been justly compared in many respects with musical figures, and which also shew a certain orderly arrangement, constantly employs curved lines and not lines broken into determinate lengths. But in the first place the art of arabesques really began with the Greek meander, which is composed of straight lines set at right angles to each other, following at exactly equal lengths, and cutting one another off in degrees. In the second place, the eye which contemplates arabesques can take in and compare all parts of the curved lines at once, and can glance to and fro, and return to its first contemplation. Hence, notwithstanding the continuous curvature of the lines, their paths are perfectly comprehensible, and it became possible to renounce the strict regularity of the Grecian arabesques in favour of the curvilinear freedom. But whilst freer forms are thus admitted for individual small decorations in architecture, the division of any great whole, whether it be a series of arabesques or a row of windows or columns, &c., throughout a building, is still tied down to the simple arithmetical law of repetition of similar parts at equal intervals.

The individual parts of a melody reach the ear in succession. We cannot perceive them all at once. We cannot observe backwards and forwards at pleasure. Hence for a clear and sure measurement of the change of pitch, no means were left but progression by determinate degrees. This series of degrees is laid down in the musical scale. When the wind howls and its pitch rises or falls in insensible gradations without any break, we have nothing to measure the variations of pitch, nothing by which we can compare the later with the earlier sounds, and comprehend the extent of the change. The whole phenomenon produces a confused, unpleasant impression. The musical scale is as it were the divided rod, by which we measure progression in pitch, as rhythm measures progression in time. Hence the analogy between the scale of tones and rhythm naturally occurred to musical theoreticians of ancient as well as modern times.

We consequently find the most complete agreement among all nations that use music at all, from the earliest to the latest times, as to the separation of certain determinate degrees of tone from the possible mass of continuous gradations of sound, all of which are audible, and these degrees form the scale in which the melody moves. But in selecting the particular degrees of pitch, deviations of national taste become immediately apparent. The number of scales used by different nations and at different times is by no means small.

Let us inquire, then, what motive there can be for selecting one tone rather than another in its neighbourhood for the step succeeding any given tone. We remember that in sounding two tones together such a relation was observed. We found that under such circumstances certain particular intervals, namely the consonances, were distinguished from all other intervals which were nearly the same, by the absence of beats. Now some of these intervals, the Octave, Fifth, and Fourth, are found in all the musical scales known. [3] Recent theoreticians that have been born and bred in the system of harmonic music, have consequently supposed that they could explain the origin of the scales, by the assumption that all melodies arise from thinking of a harmony to them, and that the scale itself, considered as a melody of the key, arose from resolving the fundamental chords of the key into their separate tones. This view is certainly correct for modern scales; at least these have been modified to suit the requirements of the harmony. But scales existed long before there was any knowledge or experience of harmony at all. And when we see historically what a long period of time musicians required to learn how to accompany a melody by harmonies, and how awkward their first attempts were, we cannot feel a doubt that ancient composers had no feeling at all for harmonic accompaniment, just as even at the present day many of the more gifted Orientals are opposed to our own harmonic music. We must also not forget that many popular melodies, of older times or foreign origin, scarcely admit of any harmonic accompaniment at all, without injury to their character.

The same remark applies to Rameau’s assumption of an 'understood' fundamental bass in the construction of melodies or scales for a single voice. A modern composer would certainly imagine to himself at once the fundamental bass to the melody he invents. But how could that be the case with musicians who had never heard any harmonic music, and had no idea how to compose any? Granted that an artist's genius often unconsciously 'feels out' many relations, we should be imputing too much to it if we asserted that the artist could observe relations of tones which he had never or very rarely heard, and which were destined not to be discovered and employed till many centuries after his time.

It is clear that in the period of homophonic music, the scale could not have been constructed so as to suit the requirements of chordal connections unconsciously supplied. Yet a meaning may be assigned, in a somewhat altered form, to the views and hypotheses of musicians above mentioned, by supposing that the same physical and physiological relations of the tones, which become sensible when they are sounded together and determine the magnitude of the consonant intervals, might also have had an effect in the construction of the scale, although under somewhat different circumstances.

Let us begin with the Octave, in which the relationship to the fundamental tone is most remarkable. Let any melody be executed on any instrument which has a good musical quality of tone, such as a human voice; the hearer must have heard not only the primes of the compound tones, but also their upper Octaves, and, less strongly, the remaining upper partials. When, then, a higher voice afterwards executes the same melody an Octave higher, we hear again a part of what we heard before, namely the evenly numbered partial tones (p. 49) of the former compound tones, and at the same time we hear nothing that we had not previously heard. Hence the repetition of a melody in the higher Octave is a real repetition of what has been previously heard, not of all of it, but of a part.[4] If we allow a low voice to be accompanied by a higher in the Octave above it, the only part music which the Greeks employed, we add nothing new, we merely reinforce the evenly numbered partials. In this sense, then, the compound tones of an Octave above are really repetitions of the tones of the lower Octaves, or at least of part of their constituents. Hence the first and chief division of our musical scale is that into a series of Octaves. In reference to both melody and harmony, we assume tones of different Octaves which bear the same name, to have the same value, and, in the sense intended, and up to a certain point, this assumption is correct. An accompaniment of Octaves gives perfect consonance, but it gives nothing additional; it merely reinforces tones already present. Hence it is musically applicable for increasing the power of a melody which has to be brought out strongly, but it has none of the variety of polyphonic music, and therefore is felt to be monotonous, and it is consequently forbidden in polyphonic music.

What is true of the Octave is true in a less degree for the Twelfth. If a melody is repeated in the Twelfth we again hear only what we had already heard, but the repeated part of what we heard is much weaker, because only the third, sixth, ninth, &c., partial tone is repeated, whereas for repetition in the Octave, instead of the third partial, the much stronger second and weaker fourth partial is heard, and in place of the ninth, the eighth and tenth occur, &c. Hence repetition of a melody in the Twelfth is less complete than repetition in the Octave, because only a smaller part of what had been already heard is repeated. In place of this repetition in the Twelfth, we may substitute one an Octave lower, namely in the Fifth. Repetition in the Fifth is not a pure repetition, as that in the Twelfth is. Taking 2 for the pitch number of the prime tone, the partials are (197c, d)

$$ \begin{array}{lllllllllllll} \text{for the fundamental compound}& & 2 & & 4 & & 6 & & 8 & & 10 & & 12 \\ \text{for the Twelfth}& & & & & & 6 & & & & & & 12 \\ \text{for the Fifth}& & & 3 & & & 6 & & & 9 & & & 12 \end{array} $$

When we strike the Twelfth we repeat the simple tones 6 and 12, which already existed in the fundamental compound tone. When we strike the Fifth, we continue to repeat the same simple tones, but we also add two others, 3 and 9. Hence for the repetition in the Fifth, only a part of the new sound is identical with a part of what had been heard, but it is nevertheless the most perfect repetition which can be executed at a smaller interval than an Octave. This is clearly the reason why unpractised singers, when they wish to join in the chorus to a song that does not suit the compass of their voice, often take a Fifth to it. This is also a very evident proof that the uncultivated ear regards repetition in the Fifth as natural. Such an accompaniment in the Fifth and Fourth is said to have been systematically developed in the early part of the middle ages. Even in modern music, repetition in the Fifth plays a prominent part next to repetition in the Octave. In normal fugues the theme, as is well known, is first repeated in the Fifth; in the normal form of instrumental pieces, that of the Sonata, the theme in the first movement is transposed to the Fifth, returning in the second part to the fundamental tone. This kind of imperfect repetition of the impression in the Fifth induced the Greeks also to divide the interval of the Octave into two equivalent sections, namely two Tetrachords. Our major scale on being divided in this manner would be:—

$$ \begin{gathered} \underbrace{c \hspace{3mm} d \hspace{3mm} e \hspace{3mm} f}_{\text{I.}}\hspace{3mm}\rlap{$\overbrace{\phantom{g \hspace{3mm}a \hspace{3mm} b \hspace{3mm} c' }}^{\text{II.}}$}g \hspace{3mm} a \hspace{3mm} b\hspace{3mm} \underbrace{c' \hspace{3mm} d' \hspace{3mm} e' \hspace{3mm} f'}_{\text{III.}} \end{gathered} $$

The succession of tones in the second tetrachord is a repetition of that in the first, transposed a Fifth.[5] To pass into the Octave division, the successive tetrachords must be alternately separate and connected. They are said to be connected, or conjunct, when, as in II. and III., the last tone \(c\) of the lower becomes the first of the higher tetrachord; and separate, or disjunct, when, as in I. and II., the last tone of the lower is different from the upper. In the second tetrachord \(g\) to \(c\), every ascending series of tones necessarily leads to \(c'\) as the final tone, and this \(c'\) is also the Octave of the fundamental tone of the first tetrachord. Now this \(c'\) is the Fourth of \(g\), the fundamental tone of the second tetrachord. To make the succession of tones the same in both tetrachords, the lower tetrachord had to be increased by the tone \(f\) which answers to \(c'\). The Fourth \(f\), however, would have suggested itself in the same way as the Fifth, independently of this analogy of the tetrachords. The Fifth is a compound tone in which the second partial is the third partial of the fundamental compound tone; the Fourth is a compound tone in which the third partial is the same as the second of the Octave. Hence the limits of the two analogous divisions of the Octave are settled, namely: —

$$ \begin{gathered} c-f, \hspace{5mm} g-c', \end{gathered} $$

but the mode of filling up these gaps remains arbitrary, and different plans for doing so were adopted by the Greeks themselves at different periods, and others again by other nations. But the division of the scale into octaves, and the octave into two analogous tetrachords, occurs everywhere, almost without exception.

Boethius (De Musica, lib. i. cap. 20) informs us that according to Nicomachus the most ancient method of tuning the lyre down to the time of Orpheus, consisted of open tetrachords,

$$ \begin{gathered} c - f - g - c', \end{gathered} $$

with which certainly it was scarcely possible to construct a melody. But as it contained the chief degrees of the pitch of ordinary speech, a lyre of this kind might possibly have served to accompany declamation.

The relationship of the Fifth, and its inversion the Fourth, to the fundamental tone, is so close that it has been acknowledged in all known systems of music.[6] On the other hand, many variations occur in the choice of the intermediate tones which have to be inserted between the terminal tones of the tetrachord. The interval of a Third is by no means so clearly defined by easily appreciable partial tones, as to have forced itself from the first on the ear of unpractised musicians. We must remember that even if the fifth partial tone existed in the compound tones of the musical instruments employed, it would have had to contend with the much louder prime tone, and would also have been covered by the three adjacent and lower partials. As a matter of fact, the history of musical systems shews that there was much and long hesitation as to the tuning of the Thirds. And the doubt is even yet felt when Thirds are used in pure melody, unconnected with any harmonies. I must own that on observing isolated intervals of this kind, I cannot come to perfectly certain results, but I do so when I hear them in a well-constructed melody with distinct tonality. The natural major Thirds of 4 : 5 thus seem to me calmer and quieter than the sharper major Thirds of our equally tempered modern instruments, or with the still sharper major Thirds which result from the Pythagorean tuning with perfect Fifths. Both of the latter intervals have a strained effect Most of our modern musicians, accustomed to the major Thirds of the equal temperament, prefer them to the perfect major Thirds, when melody alone is concerned. But I have convinced myself that artists of the first rank, like Joachim, use the Thirds of 4 : 5 even in melody. For harmony there is no doubt at all. Every one chooses the natural major Thirds. In Chapter XVI. I shall describe an instrument which will enable any one to perform experiments of this kind. [7]

Under these circumstances another principle for determining the small intervals of the scale was resorted to during the infancy of music, and seems to be still employed among the less civilised nations. This principle, which has subsequently had to yield to that of tonal relationship, consists in an endeavour to distinguish equal intervals by ear, and thus make the differences of pitch perceptibly uniform.

This attempt has never prevailed over the feeling of tonal relationship for the division of the Fourth, at least in artistically developed music. But in the division of smaller intervals we shall find it applied as an auxiliary in many of the less usual divisions of the Greek tetrachord and in the scales of Oriental nations. But arbitrary divisions which are independent of tonal relationship, disappeared everywhere in exact proportion to the higher development of the musical art.

We will now inquire what kind of a scale we should obtain by pursuing to its consequences the natural relationship of the tones. We shall consider musical tones to be related in the first degree which have two identical partial tones; and related in the second degree, when they are both related in the first degree to some third musical tone. The louder the coincident in proportion to the non-coincident partials of compound tones related in the first degree, the closer is their relationship and the more easily will both singers and hearers feel the common character of both the tones. Hence it follows that the feeling for tonal relationship ought to differ with the qualities of tone: and I believe that this states a fact in nature, because flutes and the soft stops of organs, on which chords are somewhat colourless owing to an absence of upper partials and a consequent incomplete definition of dissonances, retain much of the same colourless character in melodies. This, I think, depends upon the fact, that, for such qualities of tone, the recognition of the natural intervals of the Thirds and Sixths, and perhaps even of the Fourths and Fifths, does not result from the immediate sensation of the hearer, but at most from his recollection. When he knows that on other instruments and in singing he has been able by immediate sensation to recognise the Thirds and Sixths as naturally related tones, he acknowledges them as well-known intervals even when executed by a flute or on the soft stops of an organ. But the mere recollection of an impression cannot possibly have the same freshness and power as the immediate sensation itself.

Since the closeness of relationship depends on the loudness of the coincident upper partial tones, and those having a higher ordinal number are usually weaker than those having a lower one, the relationship of two tones is generally weaker, the greater the ordinal number of the coincident partials. These ordinal numbers, as the reader will recollect from the theory of consonant intervals, also give the ratio of the vibrational numbers of the corresponding notes.

In the following table, the first horizontal line contains the ordinal numbers of the partial tones of the tonic \(c\), and the first vertical column those of the corresponding tone in the scale. Where the corresponding vertical columns and horizontal lines intersect, the name of the tone of the scale is given for which this coincidence holds. Only such notes are admitted as are distant from the tonic by less than an Octave. Below each degree of the scale are placed the two ordinal numbers of the coincident partials, which will serve as a scale for measuring the closeness of the relationship.

Partial Tones of the Tonic
1 2 3 4 5 6
1 \(c\)
2 \(C\)
3 \(F\)
4 \(C\)
5 \(E\flat\)
6 \(C\)

In this systematic comparison we find the following series of notes lying in the octave above the fundamental note c, and related to the tonic c in the first degree, arranged in the order of their relationship:

$$ \begin{array}{lllllll} &\hspace{2mm}c &\hspace{2mm}c' &\hspace{2mm}g &\hspace{2mm}f &\hspace{2mm}a &\hspace{2mm}e &\hspace{2mm}e\flat \\ &1:1 &1:2 &2:3 &3:4 &3:5 &4:5 &5:6 \end{array} $$

and the following series in the descending octave :

$$ \begin{array}{lllllll} &\hspace{2mm}c &\hspace{2mm}C &\hspace{2mm}F &\hspace{2mm}G &\hspace{3mm}E\flat &\hspace{3mm}A\flat &A \\ &1:1 &1:2 &3:2 &4:3 &5:3 &5:4 &6:5 \end{array} $$

The series is discontinued when the resultant intervals become very close. Intervals adapted for practical use must not be too close to be easily taken and distinguished. What is the smallest interval admissible in a scale is a question which different nations have answered differently according to the different direction of their taste, and perhaps also according to the different delicacy of their ear.

It seems that in the first stages of the development of music many nations avoided the use of intervals of less than a Tone, and hence formed scales, which alternated in intervals from a Tone to a Tone and a half. According to examples collected by M. Fétis,[8] a scale of this kind is found not only among the Chinese but also among the other branches of the Mongol race, among the Malays of Java and Sumatra, the inhabitants of Hudson's Bay, the Papuas of New Guinea, the inhabitants of New Caledonia, and the Fullah negroes.[9] The five-stringed lyre (Kissar) of the inhabitants of North Africa and Abyssinia, which is represented in the bas-reliefs of the Assyrian palaces as an instrument played on by captives, was also, according to Villoteau, [10] tuned by the scale of five degrees:

$$ g \;\text{\textemdash} \;a \;\text{\textemdash} \;b\; \text{\textemdash}\; d'\; \text{\textemdash} \;e' $$

Traces of an ancient scale of this kind are clearly furnished by the five-stringed lyre or lute (κιθάρα) of the Greeks. At least Terpander (circa B.C. 700-650), who played a conspicuous part in the development of ancient Greek music, and who added a seventh string to the former Cithara of six strings, used a scale composed of a tetrachord and a trichord, having the compass of an Octave and tuned thus:—

$$ \begin{gathered} e \smile f \;\text{\textemdash} \;g \;\text{\textemdash} \;a \;\text{\textemdash} \;b \smile \text{\textemdash} \;d' \;\text{\textemdash} \;e' \end{gathered} $$

in which there is no \(c'\), and the upper tetrachord has no interval of a Semitone, although there is an interval of this kind in the lower.[13]

Olympos (circa B.C. 660-620), who introduced Asiatic flute music into Greece and adapted it to Greek tastes, transformed the Greek Doric scale into one of five tones, the old enharmonic scale

$$ \begin{gathered} B \smile c \;\text{\textemdash}\; \text{\textemdash} \;e \smile f \;\text{\textemdash}\; \text{\textemdash}\;a \end{gathered} $$

This seems to indicate that he brought a scale of five tones with him from Asia, and merely borrowed the use of the intervals of a Semitone from the Greeks. Among the more cultivated nations, the Chinese and the Celts of Scotland and Ireland still retain the scale of five notes without Semitones, although both have also become acquainted with the complete scale of seven notes.

Among the Chinese, a certain prince Tsay-yu is said to have introduced the scale of seven notes amid great opposition from conservative musicians. The division of the Octave into twelve Semitones, and the transposition of scales have also been discovered by this intelligent and skilful nation. But the melodies transcribed by travellers mostly belong to the scale of five notes. The Gaels and Erse have likewise become acquainted with the diatonic scale of seven tones by means of psalmody, and in the present form of their popular melodies the missing tones are sometimes just touched as appoggiature or passing notes. These are, however, in many cases merely modern improvements, as may be seen on comparing the older forms of the melodies, and it is usually possible to omit the notes which do not belong to the scale of five tones without impairing the melody. This is not only true of the older melodies, but of more modem popular airs which were composed during the last two centuries, whether by learned or unlearned musicians. Hence the Gaels as well as the Chinese, notwithstanding their acquaintance with the modem tonal system, hold fast by the old. [15]And it cannot be denied that by avoiding the Semitones of the diatonic scale, Scotch airs receive a peculiarly bright and mobile character, although we cannot say as much for the Chinese. Both Gaels and Chinese make up for the small number of tones within the Octave by great compass of voice.[16]

The scale of five tones admits of a certain variety in its construction. Assume \(c\) as the tonic and add to it the nearest related notes in the ascending Octave, till you come to a Semitone. This gives

$$ c \;\text{\textemdash} \;c' \;\text{\textemdash} \;g \;\text{\textemdash} \;f \;\text{\textemdash} \;a $$

The next note \(e\) would form a Semitone with \(f\). In the descending Octave we find in the same way

$$ c \;\text{\textemdash} \;C \;\text{\textemdash} \;F\; \text{\textemdash} \;G \;\text{\textemdash} \;E \flat $$

The great gaps in the scales between \(c\) and \(f\) in the first, and between \(G\) and \(c\) in the second are filled up by tones related in the second degree. Since the tones related to the Octave can only be repetitions of those directly related to the tonic, the next tones to be considered are those related to the upper Fifth \(g\), and lower Fifth \(F\), and these are \(d\) (the Fifth above the upper Fifth \(g\)) and \(B \flat\), (the Fifth below the lower Fifth \(F\)). We thus obtain the scales [17]

$$ \begin{alignedat}{15} &1) &\text{Ascending} \hspace{3mm} &c \hspace{1mm}\text{\textemdash}\; &&d \hspace{1mm}\text{\textemdash} \smile\; &&f \hspace{1mm}\text{\textemdash}\; &&g \hspace{1mm}\text{\textemdash}\; &&a \hspace{1mm}\text{\textemdash} \smile\; &&c'\\ &&&\small{\text{1}} &&\small{\tfrac98} && \small{\tfrac43} && \small{\tfrac32} && \small{\tfrac53} && \small{\text{2}} \end{alignedat} $$ $$ \begin{alignedat}{20} &2) &\text{Descending} \hspace{3mm} &C \hspace{1mm}\text{\textemdash} \smile\; && E\flat \hspace{1mm}\text{\textemdash}\; &&F \hspace{1mm}\text{\textemdash}\; &&G \hspace{1mm}\text{\textemdash} \smile\; &&B\flat \hspace{1mm}\text{\textemdash}\; &&c\\ &&&\small{\text{1}} &&\small{\tfrac65} &&\small{\tfrac43} && \small{\tfrac32} && \small{\tfrac{16}{9}} && \small{\text{2}} \end{alignedat} $$

But in place of the tones more distantly related to the tonic in the first degree, both systems of tones related in the second degree might be used, and this would give a scale resulting from a simple progression by Fifths, as

$$ \begin{alignedat}{20} &3) \hspace{3mm}&&c \hspace{1mm}\text{\textemdash}\; &&d \hspace{1mm}\text{\textemdash} \smile\; &&f \hspace{1mm}\text{\textemdash}\; &&g \hspace{1mm}\text{\textemdash} \smile\; &&b\flat \hspace{1mm}\text{\textemdash}\; &&c'\\ &&&\small{\text{1}} && \small{\tfrac98} && \small{\tfrac43} && \small{\tfrac32} && \small{\tfrac{16}{9}} && \small{\text{2}} \end{alignedat} $$

Then there are also some more irregular forms of this scale of five tones, in which the major Third \(e\) replaces the Fourth \(f\), which is more nearly related to the tonic \(c\). This transformation is probably due to the modern preference for the major mode, and it has made its appearance in very many Scotch melodies. The scale is then

$$ \begin{alignedat}{20} &4) \hspace{3mm}&&c \hspace{1mm}\text{\textemdash}\; &&d \hspace{1mm}\text{\textemdash}\; &&e \hspace{1mm}\text{\textemdash} \smile\; &&g \hspace{1mm}\text{\textemdash}\; &&a \hspace{1mm}\text{\textemdash} \smile\; &&c'\\ &&&\small{\text{1}} && \small{\tfrac98} && \small{\tfrac54} && \small{\tfrac32} && \small{\tfrac53} &&\small{\text{2}} \end{alignedat} $$

The examples of a similar exchange of the Fifth \(g\) for the minor Sixth \(a\flat\) are doubtful. This would give the scale

$$ \begin{alignedat}{20} &5) \hspace{3mm} &&C \hspace{1mm}\text{\textemdash} \smile\; && E\flat \hspace{1mm}\text{\textemdash}\; &&F \hspace{1mm}\text{\textemdash} \smile\; &&A\flat \hspace{1mm}\text{\textemdash}\; &&B\flat \hspace{1mm}\text{\textemdash}\; &&c\\ &&&\small{\text{1}} &&\small{\tfrac65} &&\small{\tfrac43} && \small{\tfrac85} && \small{\tfrac{16}{9}} && \small{\text{2}} \end{alignedat} $$

The scale $$ \begin{alignedat}{20} &6) \hspace{3mm} &&c \hspace{1mm}\text{\textemdash} \smile\; && e\flat \hspace{1mm}\text{\textemdash}\; &&f\hspace{1mm} \text{\textemdash}\; &&g \hspace{1mm}\text{\textemdash}\; &&a \hspace{1mm} \text{\textemdash} \smile\; &&c\\ &&&\small{\text{1}} &&\small{\tfrac65} &&\small{\tfrac43} && \small{\tfrac32} && \small{\tfrac53} && \small{\text{2}} \end{alignedat} $$

in which all the notes are related in the first degree, but for which the nearest notes to the tonic, either way, are a Tone and a Semitone distant from it, has not yet been discovered in actual use.

The above five forms of the scale of five tones can all be so transposed that they can be played on the black notes of a piano without touching the white ones.[18] This is the well-known simple rule for composing Scotch melodies. [19] Any one of the five black notes may then be used as a tonic, but the \(B\flat\) or \(A\sharp\) having no Fifth (\(F\) or \(E\sharp\)) among the black notes has a very doubtful effect as a tonic.

The following are examples of the use of these various scales of five tones: —

1. The First Scale without Third or Seventh. Chinese, after John Barrow.[20]

2. To the Second Scale, without Second or Sixth, belong most Scotch airs which have a minor character. In the modern forms of these airs one or other of the missing tones is often transiently touched. Here follows an older form of the air called Cockle Shells:[21]

3. For the Third Scale, without Third and Sixth. Gaelic. Probably an old bagpipe tune.[22]

4. To the Fourth Scale, without Fourth or Seventh, belong most Scotch airs which have the character of a major mode. Since dozens of Scotch tunes of this kind are to be found in every collection, and are perfectly well known, I give here a Chinese temple hymn, after Bitschurin,[23] as an example:

5. For the Fifth Scale, without Second and Fifth, I have found no perfectly pure examples. But there are melodies with either only the Fifth or else with a mere transient use of both Second and Fifth. In the latter case the minor Second is used, giving it the character of the ecclesiastical Phrygian tone, for example in the very beautiful air, Auld Robin Gray. I give an example with the tonic \(f\sharp\), in which the Second (\(g\sharp\) or \(g\)) is altogether absent, and the fifth \(c\sharp\) is only once transiently touched, so that it might just as well have been omitted.

We might also in this example assume \(b\) as the tonic, and regard the conclusions as formed upon the dominant and subdominant in the old-fashioned way.[24] In these scales of five tones the determination of the tonic is much more doubtful than in the scales of seven tones.

The rule usually given for the Gaelic and Chinese scales, to omit the Fourth and Seventh, applies therefore only to the fourth of the above scales, which corresponds to our major scale. True this scale is often used in the usual Scotch airs of the present day, and is probably due to the reaction of our modern tonal system. But the examples here adduced shew that every possible position may be assumed by the tonic in the scale of five tones, if indeed we allow these scales to have a tonic at all. In Scotch melodies the omissions in both major and minor scales are so contrived as to avoid the intervals of a Semitone, and substitute for them intervals of a Tone and a half. Among the Chinese airs, however, I have found one which belongs rather to the old Greek enharmonic system, to be considered presently, and it will be explained at the same time (p. 265).

We now proceed to the construction of scales with seven degrees. The first form was developed in Greece under the influence of the tetrachordal divisions. The ancient Greek melodies had a small compass and few degrees, a peculiarity especially emphasised even by later authors, as Plutarch, but it is also found among most nations in the early stages of their musical cultivation. Hence the scale was at first formed within a less compass than an Octave, namely within the tetrachord. On looking within this compass for the tones nearest related to the limiting tonic (μέση), we find only the Thirds. Thus if we assume \(e\) (the last tone in the tetrachord, \(b - e\)) as a tonic, its next related tone within the compass of that tetrachord is \(c\), the major Third below \(e\). This gives: —

1. The ancient enharmonic tetrachord of Olympos —

$$ \begin{alignedat}{15} &b \smile\; &&c \hspace{1mm} \text{\textemdash}\hspace{1mm}\text{\textemdash}\; &&e \\ &\small{\tfrac34} &&\small{\tfrac45} &&\small{\text{1}} \end{alignedat} $$

Archytas was the first to settle that the tuning of \(c : e\) must be 4 : 5 in the enharmonic mode. The next most closely related tone to \(e\) would be the minor Third below it. Adding this we obtain:

2. The older chromatic tetrachord of the Greeks —

$$ \begin{alignedat}{15} &b \smile\; &&c \smile\; &&c\sharp \hspace{1mm} \text{\textemdash} \smile\; &&e \\ &\small{\tfrac34} &&\small{\tfrac45} &&\small{\tfrac56} &&\small{\text{1}} \end{alignedat} $$

The method of tuning the intervals here assigned agrees with the data of Eratosthenes (in the third century before Christ). The interval between \(c\) and \(c\sharp\) in this case corresponds to the small ratio \(\tfrac{25}{24}\) [= 70 cents], which is less than the Semitone \(\tfrac{16}{15}\)[= 112 cents]. Next to it comes the much wider interval, \(c\sharp - e\), corresponding to a minor Third. We should obtain a more even distribution of intervals, by measuring the minor Third upwards from the lowest tone of the tetrachord. This gives rise to

3. The diatonic tetrachord

$$ \begin{alignedat}{15} &b \smile\; &&c \hspace{1mm} \text{\textemdash}\; &&d \hspace{1mm} \text{\textemdash}\; &&e \\ &\small{\tfrac34} &&\small{\tfrac45} &&\small{\tfrac{9}{10}} &&\small{\text{1}} \end{alignedat} $$

This is the tuning assigned by Ptolemy for the diatonic tetrachord. Here we must observe that if \(e\) continued to be regarded as the tonic, \(d\) would have only a distant relation with it in the second degree through the auxiliary tone \(b\). If two tetrachords had been connected, as was very early done, thus:

$$ b \hspace{1mm}\text{\textemdash}\hspace{1mm}\text{\textemdash}\hspace{1mm} e \hspace{1mm}\text{\textemdash}\hspace{1mm} \text{\textemdash}\hspace{1mm} a $$

a closer connection in the second degree between \(d\) and \(e\) might have been obtained by tuning \(d\) as a Fifth below \(a\). Taking \(e\) as 1, \(a\) will be \(\tfrac43\), and the Fifth below it is \(d = \tfrac89\). We thus obtain the tetrachord

4. $$ \begin{alignedat}{15} &b \smile\; &&c \hspace{1mm} \text{\textemdash}\; &&d \hspace{1mm} \text{\textemdash}\; &&e \\ &\small{\tfrac34} && \small{\tfrac45} && \small{\tfrac89} && \small{\text{1}} \end{alignedat} $$

which agrees with the tuning assigned by Didymus (in the first century before Christ).

According to the old theory of Pythagoras, which will be examined presently, all the intervals of the diatonic scale should be tuned by means of intervals of a Fifth, giving:

5. $$ \begin{gathered} b \smile c \hspace{1mm} \text{\textemdash} \hspace{1mm} d \hspace{1mm} \text{\textemdash} \hspace{1mm} e\\ \hspace{0mm} \underbrace{\tfrac43 \hspace{4.5mm} \tfrac{81}{64}}_{\tfrac{243}{256}} \underbrace{\hspace{3mm} \tfrac98}_{\tfrac98}\underbrace{\hspace{3mm} \phantom{\tfrac98}1}_{\tfrac98} \end{gathered} $$

The tetrachord thus obtained is the Greek Doric, which is considered as normal, and made the basis of all considerations on other scales. Accordingly those tones which formed the lower tones of the semitonic intervals of the scale, were, at least theoretically, considered as the immovable limiting tones of the tetrachord while the intermediate tones might change their position. Practically the intonation of even these fixed tones was a little changed, as Plutarch tells us, which may mean that in the Lydian, and Phrygian modes, &c., the tonic is not selected from one of these so-called fixed tones of the tetrachords. Thus we shall see further on, that when \(d\) is the tonic, the \(b\) in the natural intonation of such a scale does not form a perfect Fifth with \(e\).

The tetrachord could, however, be differently completed by inserting tones which formed major or minor Thirds with either of the extreme tones.

Two minor Thirds give the Phrygian tetrachord —

6. $$ \begin{alignedat}{15} &d \hspace{1mm} \text{\textemdash}\; &&e \smile\; &&f \hspace{1mm} \text{\textemdash}\; &&g \\ &\small{\tfrac34} &&\small{\tfrac56} &&\small{\tfrac{9}{10}} &&\small{\text{1}} \end{alignedat} $$

If a major Third were taken upwards from the lower extreme tone, and a minor Third downwards from the upper extreme, we should obtain the Lydian Tetrachord

7. $$ \begin{alignedat}{15} &c \hspace{1mm} \text{\textemdash}\; &&d \hspace{1mm} \text{\textemdash}\; &&e \smile\; &&f \\ &\small{\tfrac34} && \small{\tfrac56} && \small{\tfrac{15}{16}} && \small{\text{1}} \end{alignedat} $$

8. Two major Thirds, as in \(b\smile c \space \text{\textemdash} \space d\sharp\smile e\), would form a variety of the chromatic scale, which does not seem to have been used, or at any rate not to have been distinguished from the chromatic form.[25]

These are all the normal subdivisions of the tetrachord that have been used. But other subdivisions occur which the Greeks themselves termed irrational (ἄλογα),[26] and we do not know with certainty how far they were practically used. One of them, the soft diatonic mode, makes use of the interval 6 : 7, which is at any rate very near to a natural consonance, being that between the Fifth and the subminor Seventh of the fundamental note, an interval occasionally used in harmonic music when unaccompanied singers take the minor Seventh of the chord of the dominant Seventh. The intervals [27]are:

9. $$ \overbrace{\underbrace{\tfrac{21}{20} \hspace{2cm} \tfrac{10}{9}}_{\large{6:7}} \hspace{2cm} \tfrac{8}{7}}^{\large{3:4}} $$

By lowering the Lichanos the Parhypatē is also flattened. However, the small interval \(\small{\tfrac{21}{20}}\) is very nearly the Pythagorean Semitone, which expressed approximately is \(\small{\frac{20}{19}}\).

The equal diatonic mode of Ptolemy, which was divided thus:[28]

10. $$ \overbrace{\underbrace{\tfrac{12}{11} \hspace{2cm} \tfrac{11}{10}}_{\large{5:6}} \hspace{2cm} \tfrac{10}{9}}^{\large{3:4}} $$

contained a perfect minor Third divided as evenly as possible.

There is a similar succession of tones, in an inverse order, in the modern Arabic scale as measured by the Syrian, Michael Meshāqah.[29] In this case the Octave is divided into twenty-four Quartertones[30] (a turned \(\flat\), standing for q the initial of quarter) to represent an added Quartertone, \(\sharp\) being two Quartertones, and \(\sharp\)\(\flat\) three Quartertones, thus ascending \(c \;c\)\(\flat\)\(\; c\sharp \;c\sharp\)\(\flat\) \(\;d \) or descending \(d \;d\flat\) \(\flat\) \(\;d\flat\;c\)\(\flat\)\(\;c\), using \(d\flat\) as the equivalent of \(c\sharp\). Then the principal scale of Meshāqah (see App. XX. sect. K.) is \(a \; 200 \; b \; 150 \; c'\)\(\flat\) \(\; 150 \; d' \; 200 \; e' \; 150 \; f'\)\(\flat\)\(\;150 \; g' \; 200 \; a'.\) Hence the tetrachord \(a : d'\), which represents 10 in the text, has one interval of 200 cents or 4 Quartertones, and two of 150 cents or 3 Quartertones. This interval of 3 Quartertones represents the trumpet intervals \(^{11}f : g = 11 : 12 = 151\) cents, and \(e : ^{11}f = 10 : 11 = 165 \) cents , and was introduced into Arabia by the lutist Zalzal, who died about 1000 years ago, and is much used in the East. — Translator.]; the tetrachord 10 has ten of them, its lowest interval four, and each of the upper intervals three. Under these circumstances the two upper intervals together form very nearly a minor Third, which, as in the equal diatonic scale of the Greeks, is divided into two equal intervals, without paying regard to any sensible relationship of the intermediate tone thus produced.

The closer the interval, the more easy and certain is its division into two intervals, by the mere feeling for difference of pitch. This is, in particular, possible for intervals which approach to the limits at which differences of pitch are distinguishable by the ear. The distinctness with which the yet sensible difference can be felt then furnishes a measure of its magnitude. In this sense we have probably to explain the possibility of the later enharmonic mode of the Greeks, which, however, had already fallen into disuse in the time of Aristoxenus, and was perhaps hunted up again by later writers as an antiquarian curiosity. In this mode the Semitone of the ancient enharmonic mode already mentioned (No. 1, p. 262) was again subdivided into two Quartertones, so that a tetrachord was produced like the chromatic one, but with closer intervals between the adjacent tones. The division of this enharmonic tetrachord [31] for the approximate Quartertone he would have about, \(c \; 56 \; c\)\(\flat\)\(\; 56 \; d'\flat \; 386 \; f\), or something sufficiently like it. Meshaqāh’s \(c \;50 \;c\)\(\flat\)\(\; 50 \; c\sharp \; 400 \; f \) would doubtless have been near enough. Probably no two lyrists tuned alike. My experience of tuning by ear is quite against any approach to the accuracy which the figures in the text would imply. — Translator.] was

11. $$ \overbrace{\tfrac{32}{31} \hspace{2cm} \tfrac{31}{30} \hspace{2cm} \tfrac{5}{4}}^{\large{3:4}} $$

This Quartertone can only be considered as a transition in the melodic movement towards the lowest extreme of the tetrachord. A similar interval occurs in this way in existing Oriental music. A distinguished musician whom I requested to pay attention to it on a visit to Cairo, wrote to me as follows: 'This evening I have been listening attentively to the song on the minarets, to try to appreciate the Quartertones, which I had not supposed to exist, as I had thought that the Arabs sang out of tune. But to-day as I was with the dervishes I became certain that such Quartertones existed, and for the following reasons. Many passages in litanies of this kind end with a tone which was at first the Quartertone and then ended in the pure tone.[32] As the passage was frequently repeated, I was able to observe this every time, and I found the intonation invariable.' The Greek writers on music themselves say that it is difficult to distinguish the enharmonic Quartertones.[33]

The later interpreters of Greek musical theory have mostly advanced the opinion that the above-mentioned differences, which the Greeks called colourings (χρόαι), were merely speculative and never came into practical use.[34] They consider that these distinctions were too delicate to produce any esthetic effect except on an incredibly well cultivated ear. But it seems to me that this opinion could never have been entertained or advanced by modern theorists, if any of them had practically attempted to form these various tonal modes and to compare them by ear. On an harmonium which will shortly be described I am able to compare natural intonation with Pythagorean, and to play the diatonic mode at one time after the method of Didymus and at another after that of Ptolemy, and also to make other deviations. It is not at all difficult to distinguish the difference of a comma \(\tfrac{81}{80}\) in the intonation of the different degrees of the scale, when well-known melodies are performed in different 'colourings,' and every musician with whom I have made the experiment has immediately heard the difference. Melodic passages with Pythagorean Thirds have a strained and restless effect, while the natural Thirds make them quiet and soft, although our ears are habituated to the Thirds of the equal temperament, which are nearer to the Pythagorean than to the natural intervals. Of course where delicacy in any artistic observations made with the senses, comes into consideration, moderns must look upon the Greeks in general as unsurpassed masters. And in this particular case they had very good reason and abundance of opportunity for cultivating their ear better than ours. From youth upwards we are accustomed to accommodate our ears to the inaccuracies of equal temperament, and the whole of the former variety of tonal modes, with their different expression, has reduced itself to such an easily apprehended difference as that between major and minor. But the varied gradations of expressions which moderns attain by harmony and modulation, had to be effected by the Greeks and other nations that use homophonic music, by a more delicate and varied gradation of the tonal modes. Can we be surprised, then, if their ear became much more finely cultivated for differences of this kind than it is possible for ours to be?

The Greek scale was soon extended to an octave. Pythagoras is said to have been the first to establish the eight complete degrees of the diatonic scale. At first two tetrachords were connected in such a way as to have a common tone, the μέση:

$$ \begin{gathered} \rlap{$\underbrace{\phantom{e \smile f \; \text{\textemdash} \; g \; \text{\textemdash} \; a }}$}e \smile f \; \text{\textemdash} \; g \; \text{\textemdash} \; \overbrace{a \smile b\flat \text{\textemdash} \;c' \; \text{\textemdash} \; d'} \end{gathered} $$

which produced a scale of seven degrees. Then this scale was changed into the following form:

$$ \begin{gathered} \underbrace{e \smile f \; \text{\textemdash} \; g \; \text{\textemdash} \; a} \; \text{\textemdash} \; \overbrace{b \smile \text{\textemdash} \; d' \; \text{\textemdash} \; e'} \end{gathered} $$

and thus made to consist of a tetrachord and a trichord, of which mention has already been made (p. 257). Finally Lichaon of Samos (according to Boethius), or Pythagoras (according to Nicomachus), completed the trichord into a tetrachord, and thus established a scale consisting of two disjunct tetrachords.

The diatonic scale thus obtained could be continued either way at pleasure by adding higher and lower octaves, and it then produced a regularly alternating series of Tones and Semitones. But for each piece of music a portion only of this unlimited diatonic scale was employed, and the tonal systems were distinguished by the character of the portions selected.

These sectional scales might be produced in very different ways. The first practical object which necessarily forces itself on attention, as soon as an instrument with a limited number of strings, like the Greek lyre, is used for executing a piece of music, is, of course, that there should be a string for every musical tone required. This prescribes a certain series of tones which must be provided and tuned on the instrument. Now as a rule when a certain series of tones is thus prescribed as a scale for the tuning of a lyre, no question is raised as to whether a tonic is to be distinguished or not, or if so which it should be. A tolerable number of melodies may be found in which the lowest tone is the tonic: others in which an interval below the tonic is touched; and others, again, in which the Fifth or Fourth above the Octave below the tonic is used. This is the kind of difference between the authentic and plagal scales of the middle ages. In the authentic scales the deepest tone of the scale, in the plagal its Fifth below or Fourth above, was the tonic; thus: [35]

$$ \rlap{$\underbrace{\phantom{d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a}}$}d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; \overbrace{a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'} $$
$$ \rlap{$\overbrace{\phantom{d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g }}$}d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; \underbrace{g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'} $$

The scales were looked upon as composed of a Fifth and a Fourth, as the braces shew. In the authentic tone the Fifth lay below; in plagal, above. Now if we have nothing else before us but a scale of this kind, which marks out the accidental compass of a series of melodies, we can collect but little respecting the key. Such scales themselves may be fittingly termed accidental. They comprise, among others, the medieval plagal scales. On the other hand, those scales which, like the modern, are bounded at each extremity by the tonic, may be termed essential. Now practical needs clearly lead in the first place to accidental scales alone. When a lyre had to be tuned to accompany the human voice in unison, it was indispensably necessary that all the tones required should be present. There was no immediate practical need for marking the tonic of a song sung in unison, or even to become fully aware that it had a tonic at all. In modern music, where the structure of the harmony essentially depends on the tonic, the case is entirely different. Theoretical considerations on the structure of melody could alone lead to distinguishing one tone as tonic. It has been already mentioned in the preceding chapter, that Aristotle, as a writer on esthetics, has left a few notices indicating such a conception, but that the authors who have specially written on music say nothing about it.

In the best times of Greece, song was usually accompanied by an eight-stringed lyre, tuned so as to embrace an Octave of tones selected from the diatonic scale. These were the following:

       1. Lydian\( \hspace{5cm}\underbrace{c \; \text{\textemdash} \; d \; \text{\textemdash} \; e \; \text{\textemdash} \; f} \; \text{\textemdash} \; \overbrace{g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c'}\)
       2. Phrygian\( \hspace{4.6cm}\underbrace{d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g} \; \text{\textemdash} \; \overbrace{a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'}\)
       3. Doric\( \hspace{5.15cm} \underbrace{e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a} \; \text{\textemdash} \; \overbrace{b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e'}\)
       4. Hypolydian\( \hspace{4.2cm} f \; \text{\textemdash} \; \rlap{$\underbrace{\phantom{g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c'}}$}g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; \overbrace{c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f'}\)
       5. Hypophrygian (Ionic)\( \hspace{2.75cm}g \; \text{\textemdash} \; \rlap{$\underbrace{\phantom{a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'}}$}a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; \overbrace{d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f' \; \text{\textemdash} \; g'}\)
       6. Hypodoric (Eolic or Locrian)\( \hspace{1.8cm} a \; \text{\textemdash} \; \rlap{$\underbrace{\phantom{b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e'}}$}b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; \overbrace{e' \; \text{\textemdash} \; f' \; \text{\textemdash} \; g' \; \text{\textemdash} \; a'}\)
       7. Mixolydian\( \hspace{4.25cm} b \; \text{\textemdash} \; \underbrace{c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f'} \; \text{\textemdash} \; \overbrace{g' \; \text{\textemdash} \; a' \; \text{\textemdash} \; b' \; \text{\textemdash} \; (c'')}\)

Hence any one of the tones in the diatonic scale could be used as the initial or final extremity of such a tonal mode. The Lydian and Hypolydian scales contain Lydian, the Phrygian and Hypophrygian contain Phrygian, and the Doric and Hypodoric contain Doric tetrachords. In the Mixolydian two Lydian tetrachords seem to have been assumed, one of which was divided, as shewn by the braces in the above examples.[36]

The scales or tropes of the best Greek period have hitherto been considered as essential, that is the lowest tone or hypatē has been considered as the tonic. But I cannot find any definite ground for this assumption. What Aristotle says, as we have seen, makes the middle tone or mesē, function as the tonic, but yet it cannot be denied that other attributes of our tonic belong to the hypatē [37] Whatever may have been the real state of the case, whether the mesē or hypatē be regarded as the tonic, whether the scales be considered as all authentic or all plagal, it is extremely probable that the Greeks, among whom we first find the complete diatonic scale, took the liberty of using every tone of this scale as a tonic, just as we have seen that every one of the five tones forming the scales of the Chinese and Gaels occasionally functions as a tonic. The same scales are also found, probably handed down immediately by ancient tradition, in the ancient Christian ecclesiastical music.

Hence if we disregard the chromatic and enharmonic scales, and the apparently arbitrary scales of the Asiatics, none of which have shewn themselves capable of further development,[38] homophonic vocal music developed seven diatonic scales, which differ from one another in about the same way as our major and minor scales. These differences will be better appreciated by making them all begin with the same tonic \(c\).[39]

Ancient Greek Names Scales beginning with c Glarean's Ecclesiastical Names Proposed New Names[40]
Mode of the:-
1. Lydian \(c \; \text{\textemdash} \; d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c'\) Ionic First (major)
2. Ionic or Hypophrygian \(c \; \text{\textemdash} \; d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c'\) Mixolydian Fourth
3. Phrygian \(c \; \text{\textemdash} \; d \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c'\) Doric minor Seventh
4. Eolic \(c \; \text{\textemdash} \; d \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c'\) Eolic minor Third (minor)
5. Doric \(c \; \text{\textemdash} \; d\flat \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c'\) Phrygian minor Sixth
6. Mixolydian \(c \; \text{\textemdash} \; d\flat \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g\flat \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c'\) Lydian minor Second
7. Syntonolydian \(c \; \text{\textemdash} \; d \; \text{\textemdash} \; e \; \text{\textemdash} \; f\sharp \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c'\) Fifth

To assist the reader I have added the names assigned to the ecclesiastical modes by Glarean, which were wrongly distributed among the scales owing to his confusing the older tonal modes with the later (transposed) minor Greek scales, but which are more known among musicians than the proper Greek names. But I shall not use Glarean's names without expressly mentioning that they refer to an ecclesiastical mode. It would be really better to forget them altogether. The old numerical notation of Ambrose was much more suitable, but as his figures have been altered again and do not suffice for all modes, I have ventured to propose a new nomenclature in the above table, which will save the reader the trouble of memorising the systems of Greek names, of which Glarean's are certainly wrong, and the others are also perhaps not quite correctly applied. The principle of the new nomenclature is this. By 'the mode of Fourth of \(C\),' is meant a mode of which \(C\) is the tonic, but which has the same signature (or additional \(\sharp\) and \(\flat\) signs) as the major scale formed on the Fourth of the diatonic scale beginning with \(C\); that is on \(F\). The minor Seventh, minor Third, minor sixth, and minor Second must always be understood as the intervals intended in this case.[41] If the major intervals were selected the tonic would not occur in their scales. Thus 'the mode of the minor Third of \(C\)' is the scale with the tonic \(C\), having the signature of \(E\flat\) major (that is \(B\flat\), \(E\flat\), \(A\flat\)), because \(E\flat\) is the minor Third of \(C\); this is therefore \(C\) minor, at least as it is played in the descending scale. I hope the reader will have no difficulty in understanding what is meant by this notation.[42]

This was the tonal system in the best times of Greek art, up to the Macedonian empire. Airs were at first limited to a tetrachord, as is still often the case in the Roman Catholic liturgy. They were afterwards extended to an Octave. Longer scales were not necessary for singing, as the Greeks refused to employ the straining upper notes, and unmetallic deep notes of the human voice. Modern Greek songs, of which Weitzmann has made a collection,[43] have also a surprisingly small compass. If Phrynis (victor in the Panathenaic competitions, B.C. 457) added a ninth string to his cithara, the chief advantage of the arrangement was to allow of passing from one kind of scale to another.

The later Greek scale, which first occurs in Euclid’s works of the third century B.C., embraces two octaves, thus arranged:

This scheme gives first the Hypodoric [Eolic, or Locrian] scale [44] for two Octaves, and then an added tetrachord which introduces a \(b\flat\) in addition to the \(b\), and thus, in modern language, allows of modulation from the principal scale into that of the subdominant.[45]

This scale, essentially of a minor character, was transposed, and thus a new series of scales were generated that correspond with the (descending) minor scales of modern music. To these were applied the old names of the tonal nodes, by giving originally to each minor mode the name belonging to that tonal mode which was formed by the section of the minor scale which lay between the extreme tones of the Hypodoric [46] scale. According to the Greek method of representing the notes, these extreme tones would have to be written \(f...f\). Their actual pitch was probably a Third lower. Thus the minor scale of \(D\) was called Lydian, because in this scale —

$$ D \; \text{\textemdash} \; E \; \text{\textemdash} \; | \; F \; \text{\textemdash} \; G \; \text{\textemdash} \; B\flat \; \text{\textemdash} \; c \; \text{\textemdash} \; d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; | \; \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' $$

the section of the scale lying between the extreme tones \(f\) and \(f\) belonged to the Lydian tonal mode. In this way the old names of the tonal modes altered their meaning into those of tonal keys. The following table shews the correspondence of these names:—

1) Hypo-doric =\(F\) minor
2) Hypo-ionic =\(F\sharp\) minor
3) Hypo-Phrygian =\(G\) minor
4) Hypo-eolic =\(G\sharp\) minor
5) Hypo-lydian =\(A\) minor
6) Doric =\(B\flat\) minor
7) Ionic (deeper Phrygian) =\(B\) minor
8) Phrygian =\(C\) minor
9) Eolic (deeper Lydian) =\(C\sharp\) minor
10) Lydian =\(D\) minor
11) Hypo-doric (Mixo-lydian) =\(E\flat\) minor
12) Hypo-ionic (Higher Mixo-lydian) =\(E\) minor
13) Hyper-phrygian (Hyper-mixo-lydian) =\(f\) minor
14) Hyper-eolic =\(f\sharp\) minor
15) Hyper-lydian =\(g\) minor

Within each of these scales each of the previously mentioned tonal modes might be formed, by using the corresponding part of the scale. Besides this it was possible to pass into the conjunct tetrachord and thus modulate into the tonal key of the subdominant.

The experiments on transposition which formed the basis of these scales shewed that the Octave might be considered as composed approximately of twelve Semitones. Even Aristoxenus knew that by taking a series of twelve Fifths we reached a tone that was at least very near to a higher Octave of the initial tone. Thus in the series

$$ f \; \text{\textemdash} \; c \; \text{\textemdash} \; g \; \text{\textemdash} \; d \; \text{\textemdash} \; a \; \text{\textemdash} \; e \; \text{\textemdash} \; b \; \text{\textemdash} \; f\sharp \; \text{\textemdash} \; c\sharp \; \text{\textemdash} \; g\sharp \; \text{\textemdash} \; d\sharp \; \text{\textemdash} \; a\sharp \; \text{\textemdash} \; e\sharp $$

he identified \(e\sharp\) with \(f\), and by thus closing the series of tones he obtained a cycle of Fifths. Mathematicians denied the fact, and with reason, because if the Fifths are taken perfectly true, \(e\sharp\) is a little sharper than \(f\). For practical purposes, however, the error was quite insensible, and might be justly neglected in homophonic music in particular.[47]

This closes the development of the Greek tonal system. Complete as is our acquaintance with its outward form, we know but little of its real nature, because the examples of melodies which we possess are not only few in number, but very doubtful in origin.

Whatever may have been the nature of tonality in Greek scales, and however numerous may be the questions about it that are still unresolved, yet so far as the theory of the general historical development of tonal modes is concerned we learn all we want from the laws of the earliest Christian ecclesiastical music, which at its commencement touched upon the ancient construction as it died out. In the fourth century of our era, Bishop Ambrose, of Milan, established four scales for ecclesiastical song, which in the untransposed diatonic scale were:

First mode: \(d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'\) Mode of the minor Seventh
Second mode: \(e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e'\) Mode of the minor Sixth
Third mode: \(f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f'\) Mode of the Fifth (unmelodic)
Fourth mode: \(g \; \text{\textemdash} \; a \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f' \; \text{\textemdash} \; g'\) Mode of the Fourth

The variable character of the one \(b\), which was transmutable into \(b\flat\) in the later Greek scales, remained, and produced the following scales: —

First: \(d \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \; \text{\textemdash} \; d'\) Mode of the minor Third
Second: \(e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e'\) Mode of the minor Second (unmelodic)
Third: \(f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f'\) Mode of the First (major)
Fourth: \(g \; \text{\textemdash} \; a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \; \text{\textemdash} \; d' \; \text{\textemdash} \; e' \; \text{\textemdash} \; f' \; \text{\textemdash} \; g'\) Mode of the minor Seventh

There can be no doubt that these Ambrosian scales are to be regarded as essential (see p. 267), for the old rule is that melodies in the first are to end in \(d\), those in the second in \(e\), those in the third in \(f,\) and those in the fourth in \(g\), and this marks the initial tones of the scale as tonics. We may certainly assume that this arrangement was made by Ambrose for his choristers as a practical simplification of the old musical theory, which was overloaded with an inconsistent nomenclature. And this leads us to conclude that we were right in conjecturing that the similar older Greek scales could have been really used as different essential scales.

Pope Gregory the Great inserted between the Ambrosian essential scales the same number of accidental scales (p. 267), called plagal, proceeding from the Fifth to the Fifth of the tonic. The Ambrosian scales were, then, called authentic for distinction. The existence of these plagal ecclesiastical scales helped to increase the confusion which broke over the ecclesiastical scales towards the end of the middle ages, as composers began to neglect the rules which fixed the terminal tones, and this confusion assisted in favouring a freer development of the tonal system. This confusion also shewed, as we remarked in the last chapter (p. 243), that no feeling for the thorough predominance of the tonic was much developed in the middle ages. But a step, at least, was made in advance of the Greeks, by recognising as a rule that the piece should close on the tonic, although this rule was not always observed.

Glarean endeavoured in 1547 to reduce the theory of the scales to order again, in his Dodecachordon. He shewed by an examination of the musical compositions of his contemporaries, that six, and not four, authentic scales should be distinguished, and adorned them with the Greek names in the table on p. 269. Then he assumed six plagal scales, and hence on the whole distinguished twelve modes, whence the name of his book. Hence down to the sixteenth century essential and accidental scales were reckoned as parts of one series. Among Glarean's scales one is unmelodic, namely the mode of the Fifth, which he calls the Lydian. There are no examples of these to be found, as we know from a careful examination of medieval compositions made by Winterfeld,[48] and this confirms Plato’s opinion of the Mixolydian and Hypolydian modes.

Hence there remains the following five melodic tonal modes applicable strictly for homophonic and polyphonic vocal music, namely:

In our Nomenclature Ancient Greek Glarean's Names Scale
1 Major Mode Lydian Ionic \(C \; \text{\textemdash} \; c\)
2 Mode of the Fourth Ionic Mixolydian \(G \; \text{\textemdash} \; g\)
3 Mode of the minor Seventh Phrygian Doric \(D \; \text{\textemdash} \; d\)
4 Mode of the minor Third Eolic Eolic \(A \; \text{\textemdash} \; a\)
5 Mode of the minor Sixth Doric Phrygian \(E \; \text{\textemdash} \; e\)

The rational construction of these scales when extended to the Octave or beyond the Octave results from the principle of tonal relationship already explained.[49] The limits of the extent to which tones related in the first degree should be used, are determined by the necessity of avoiding intervals too close to be distinguished with certainty. The larger gaps thus left have to be filled with the tones most nearly related in the second degree.

The Chinese and Gaels made the whole Tone \(\small{\tfrac{10}{9}}\) [= 182 cents] the smallest interval.[50] The Orientals, as we have seen, still retain Quartertones. The Greeks experimented with them, but soon gave them up and kept to the Semitone \(\small{\tfrac{16}{15}}\) [ = 112 cents] as the smallest.

European nations have followed Greek habits, and retained the Semitone \(\small{\tfrac{16}{15}}\) [ = 112 cents] as the limit. The interval between \(E\flat (\small{\tfrac{6}{5}})\) [ = 316 cents] and \(E (\small{\tfrac{5}{4}})\) [ = 386 cents], and between \(A\flat (\small{\tfrac{8}{5}})\) [ = 814 cents] and \(A (\small{\tfrac{5}{3}})\) [ = 884 cents], in the natural scale is smaller, being \(\small{\tfrac{25}{24}}\) [ = 70 cents], and we consequently avoid using both \(E\flat\) and \(E\), or both \(A\flat\) and \(A\) in the same scale. We thus obtain the following two series of intervals between the most nearly related tones for ascending and descending scales:

$$ \begin{alignedat}{10} \text{Ascending:} \; \; \; &c \; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' \\ & \hspace{5.5mm} \small{\tfrac{5}{4}} \hspace{6.5mm} \small{\tfrac{16}{15}} \hspace{5mm} \small{\tfrac{9}{8}} \hspace{5mm} \small{\tfrac{10}{9}} \hspace{7mm} \small{\tfrac{6}{5}} \end{alignedat} $$ $$ \begin{alignedat}{10} \text{Descending:} \; \; \; &c \; \text{\textemdash} \; \text{\textemdash} \; A\flat \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; C \\ & \hspace{5.5mm} \small{\tfrac{5}{4}} \hspace{9mm} \small{\tfrac{16}{15}} \hspace{6mm} \small{\tfrac{9}{8}} \hspace{6mm} \small{\tfrac{10}{9}} \hspace{9mm} \small{\tfrac{6}{5}} \end{alignedat} $$

Intonation changing keyboard

Select intonation between 12 tone equal temperment and Helmholtz's suggested intonation. Harmonics are highlighted above the keys, and the Lissajous figure below displays the harmonicity of intervals.

The numbers below the series shew the intervals between the two tones between which they are placed.[51]

It is at once seen that the intervals from and to the tonic are too large, and might be further divided. But as we have come to the limits of relationship in the first degree, we have to fill these gaps by tones related in the second degree. The closest relationship in the second degree is necessarily furnished by the tones most nearly related to the tonic. Among these the Octave stands first. The tones related to the Octave of the tonic are of course the same as those related to the tonic itself; but by passing to the Octave of the tonic we obtain the descending in place of the ascending scale, and conversely.

Thus, ascending from \(c\) we found the following degrees of our major scale —

$$ c \; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' $$

But taking the tones related to \(c'\), we obtain —

$$ c \; \text{\textemdash} \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; \text{\textemdash} \; c' $$

Hence the second degree of relationship to the tonic gives an ascending minor scale. In this scale \(e\flat\) is given as the major Sixth below \(c'\). But it has also the weak relationship to \(c\) marked by 5 : 6. Now we found that the sixth partial of a compound tone was clearly audible in many qualities of tone for which the seventh or eighth could not be heard; for example, on the pianoforte, the narrower organ pipes, and the mixture stops of the organ. Hence the relationship expressed by 5 : 6 may often become evident as a natural relationship in the first degree. This, however, could scarcely be the case for the relationship \(c\; \text{\textemdash} \;a\flat\) or 5 : 8. Hence it is more natural to change \(e\) into \(e\flat\) than \(a\) into \(a\flat\) in the ascending scale. The latter, \(a\flat\), can only be related to the tonic in the second degree. The three ascending scales in order of intelligibility are, therefore — [52]

$$ c \; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' $$ $$ c \; \text{\textemdash} \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' $$ $$ c \; \text{\textemdash} \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; \text{\textemdash} \; c' $$

These distinctions based on a relationship in the second degree, through the medium of the Octave, are certainly very slight, but they make themselves felt in the well-known transformation of the ascending minor scale, to which these distinctions clearly refer.

Descending from \(c\), instead of the relations in the first degree, given in

$$ c \; \text{\textemdash} \; \text{\textemdash} \; A\flat \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; C $$

we may assume relations in the second degree, that is of the deeper \(C\), and obtain

$$ c \; \text{\textemdash} \; \text{\textemdash} \; A \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E \; \text{\textemdash} \; \text{\textemdash} \; C $$

In the latter, \(A\) is connected with the initial tone by the distant relationship in the first degree, 5 : 6, and \(E\) only by a relationship in the second degree. Hence the third descending scale

$$ c \; \text{\textemdash} \; \text{\textemdash} \; A \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; C $$

which we also found as an ascending scale, For descending scales we have therefore the following series. [53]

$$ c \; \text{\textemdash} \; \text{\textemdash} \; A\flat \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; C $$ $$ c \; \text{\textemdash} \; \text{\textemdash} \; A \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; C $$ $$ c \; \text{\textemdash} \; \text{\textemdash} \; A \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E \; \text{\textemdash} \; \text{\textemdash} \; C $$

Generally, since all Octaves of the tonic, distant or near, higher or lower are so closely related that they can be almost identified with it, all higher and lower Octaves of the individual degrees of the scale are almost as closely related to the tonic, as those of the next adjacent tonic of the same name.

Next to the relations of the Octave \(c'\) of \(c\), follow those of \(g\), the Fifth above, and \(F\) the Fifth below \(c\). We must therefore proceed to study their effect in the construction of the scale. Let us begin with the relations of \(g\), the Fifth above the tonic.[54]

$$ \begin{gathered} \text{Related to }c: c\; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' \\ \end{gathered} $$ $$ \begin{gathered} \text{Related to }g: c \;\; d \;\; e\flat \;\;\; \text{\textemdash} \; \text{\textemdash} \;\; g \; \text{\textemdash} \; \text{\textemdash} \; b \; \text{\textemdash} \; c' \end{gathered} $$

Uniting the two, we have —

1) The Major Scale (Lydian mode of the ancient Greeks): $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&d \; \text{\textemdash} \; &&e \; \text{\textemdash} \; &&f \; \text{\textemdash} \; &&g \; \text{\textemdash} \; &&a \; \text{\textemdash} \; &&b \; \text{\textemdash} \; &&c' \\ &\small{\text{1}} &&\small{\tfrac98} &&\small{\tfrac54} &&\small{\tfrac43} &&\small{\tfrac32} &&\small{\tfrac53} &&\small{\tfrac{15}{8}} &&\small{2} \end{alignedat} $$

The change of \(e\) into \(e\flat\) is here facilitated by its second relationship to \(g\). This gives —

2) The Ascending Minor Scale: $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&d \; \text{\textemdash} \; &&e\flat \; \text{\textemdash} \; &&f \; \text{\textemdash} \; &&g \; \text{\textemdash} \; &&a \; \text{\textemdash} \; &&b \; \text{\textemdash} \; &&c' \\ &\small{\text{1}} &&\small{\tfrac98} &&\small{\tfrac65} &&\small{\tfrac43} &&\small{\tfrac32} &&\small{\tfrac53} &&\small{\tfrac{15}{8}} &&\small{2} \end{alignedat} $$

$$ \begin{gathered} \text{Related to }c: c\; \text{\textemdash} \; \text{\textemdash} \; A\flat \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; C \\ \end{gathered} $$ $$ \begin{gathered} \text{Related to }g: c \; B\flat \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \;G \;\text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \;E\flat \; \text{\textemdash} \; D \; \text{\textemdash} \; C \end{gathered} $$


3) The Descending Minor Scale (Hypodoric or Eolic mode of the ancient Greeks — our mode of the minor Third): $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&B\flat \; \text{\textemdash} \; &&A\flat \; \text{\textemdash} \; &&G \; \text{\textemdash} \; &&F \; \text{\textemdash} \; &&E\flat \; \text{\textemdash} \; &&D \; \text{\textemdash} \; &&C \\ &\small{2} &&\small{\tfrac95} &&\small{\tfrac85} &&\small{\tfrac32} &&\small{\tfrac43} &&\small{\tfrac65} &&\small{\tfrac98} &&\small{\text{1}} \end{alignedat} $$

or in the mixed scale, changing \(A\flat\) into \(A\)

4) Mode of the minor Seventh (ancient Greek Phrygian): $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&B\flat \; \text{\textemdash} \; &&A \; \text{\textemdash} \; &&G \; \text{\textemdash} \; &&F \; \text{\textemdash} \; &&E\flat \; \text{\textemdash} \; &&D \; \text{\textemdash} \; &&C \\ &\small{2} &&\small{\tfrac95} &&\small{\tfrac53} &&\small{\tfrac32} &&\small{\tfrac43} &&\small{\tfrac65} &&\small{\tfrac98} &&\small{\text{1}} \end{alignedat} $$

On examining the relations of \(F\), the Fifth below the tonic \(c\), the following scales result:

$$ \begin{gathered} \text{Related to }c: c\; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; \; g \;\; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; c' \\ \end{gathered} $$ $$ \begin{gathered} \text{Related to }F: c \; \text{\textemdash} \; d \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; f \;\text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \;a \; \text{\textemdash} \; b\flat \; \text{\textemdash} \; c' \end{gathered} $$

This gives —

5) The mode of the Fourth (ancient Greek hypophrygian or Ionic): $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&d \; \text{\textemdash} \; &&e \; \text{\textemdash} \; &&f \; \text{\textemdash} \; &&g \; \text{\textemdash} \; &&a \; \text{\textemdash} \; &&b\flat \; \text{\textemdash} \; &&c' \\ &\small{\text{1}} &&\small{\tfrac{10}{9}} &&\small{\tfrac54} &&\small{\tfrac43} &&\small{\tfrac32} &&\small{\tfrac53} &&\small{\tfrac{16}{9}} &&\small{2} \end{alignedat} $$

By changing \(e\) into \(e\flat\), we again obtain —

6) The mode of the minor Seventh, but with a different determination of the intercalary tones \(d\) and \(b\flat\), from those in No. 4: $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&d \; \text{\textemdash} \; &&e\flat \; \text{\textemdash} \; &&f \; \text{\textemdash} \; &&g \; \text{\textemdash} \; &&a \; \text{\textemdash} \; &&b\flat \; \text{\textemdash} \; &&c' \\ &\small{\text{1}} &&\small{\tfrac{10}{9}} &&\small{\tfrac65} &&\small{\tfrac43} &&\small{\tfrac32} &&\small{\tfrac53} &&\small{\tfrac{16}{9}} &&\small{2} \end{alignedat} $$

$$ \begin{gathered} \text{Related to }c: c\; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; A\flat \; \text{\textemdash} \; G \; \text{\textemdash} \; F \; \text{\textemdash} \; E\flat \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; C \\ \end{gathered} $$ $$ \begin{gathered} \text{Related to }F: c \; \text{\textemdash} \; B\flat \; \text{\textemdash} \; A \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \; F \;\text{\textemdash} \; \text{\textemdash} \; \text{\textemdash} \;D\flat \; \text{\textemdash} \; C \end{gathered} $$

giving :—

7) The mode of the minor Sixth (ancient Greek Doric): $$ \begin{alignedat}{30} &c \; \text{\textemdash} \; &&B\flat \; \text{\textemdash} \; &&A\flat \; \text{\textemdash} \; &&G \; \text{\textemdash} \; &&F \; \text{\textemdash} \; &&E\flat \; \text{\textemdash} \; &&D\flat \; \text{\textemdash} \; &&C \\ &\small{2} &&\small{\tfrac{16}{9}} &&\small{\tfrac85} &&\small{\tfrac32} &&\small{\tfrac43} &&\small{\tfrac65} &&\small{\tfrac{16}{15}} &&\small{\text{1}} \end{alignedat} $$

In this way the melodic tonal modes of the ancient Greeks and Christian Church have all been rediscovered by a consistent method of derivation. As long as homophonic vocal music is alone considered, all these tonal modes are equally justified in their construction.

The scales have been given above in the order in which they are most naturally deduced. But, as we have seen, each of the three scales

$$ \begin{gathered} c \; \text{\textemdash} \; \text{\textemdash} \; e \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' \end{gathered} $$ $$ \begin{gathered} c \; \text{\textemdash} \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a \; \text{\textemdash} \; \text{\textemdash} \; c' \end{gathered} $$ $$ \begin{gathered} c \; \text{\textemdash} \; \text{\textemdash} \; e\flat \; \text{\textemdash} \; f \; \text{\textemdash} \; g \; \text{\textemdash} \; a\flat \; \text{\textemdash} \; \text{\textemdash} \; c' \end{gathered} $$

can be played either upwards or downwards, although the first is best suited to ascending and the last to descending progression, and hence the gaps of any one of them may be filled up with either the relations of \(F\) or the relations of \(g\), or even one gap with those of \(F\) and the other with those of \(g\).

The pitch numbers of the tones directly related to the tonic are of course fixed [55] and unchangeable, because they are given by the condition that the tones should form consonances with the tonic, and are thus more strictly determined than by any more distant connection. On the other hand, the intercalary tones related in the second degree are by no means so precisely fixed.

Taking \(c=1\), we have for the Second—

  1. the \(d\) derived from \(g=\small{\tfrac98}\), [=204 cents]
  2. the \(d\) derived from \(f=\small{\tfrac{10}{9}=\tfrac{80}{81}\times\tfrac98}\), [=182 cents]
  3. the \(d\flat\) derived from \(f=\small{\tfrac{16}{15}}\), [=112 cents]

and for the Seventh —

  1. the \(b\) derived from \(g=\small{\tfrac{15}{8}}\), [=1088 cents]
  2. the \(b\flat\) derived from \(g=\small{\tfrac95}\), [=1018 cents]
  3. the \(b\flat\) derived from \(f=\small{\tfrac{16}{9}=\tfrac{80}{81}\times\tfrac95}\), [=996 cents]

Hence while \(b\) and \(d\flat\) are given with certainty, \(b\flat\) and \(d\) are uncertain. Either of them may be distant from the tonic by the major Tone \(\small{\tfrac98}\) [= 204 cents] or the minor Tone \(\small{\tfrac{10}{9}}\) [= 182 cents] .

In order henceforth to mark this difference of intonation with certainty and without ambiguity, we will introduce a method of distinguishing the tones determined by a progression of Fifths, from those given by the relationship of a Third to the tonic. We have already seen that these two methods of determining the tones lead to somewhat different pitches, and hence in accurate theoretical researches both kinds of tones must be kept distinct, although in modern music they are practically confused.

The idea of this notation belongs to Hauptmann, but as the capital and small letter which he uses, and which I also, in consequence, employed in the first edition of this book, have a different meaning in our method of writing tones, I now introduce a slight modification of his notation.

Let \(C\) be the initial tone, and write [56] its Fifth \(G\), the Fifth of this Fifth \(D\), and so on. In the same way let the Fourth of \(C\) be \(F\), the Fourth of this Fourth \(B\flat\), and so on. In this way we have a series of Tones, here written with simple capitals, all distant from each other by a perfect Fifth or a perfect Fourth: [57]

$$ \begin{gathered} B\flat \; \pm \; F \; \pm \; C \; \pm \; G \; \pm \; D \; \pm \; A \; \pm \; E, \; \&c. \end{gathered} $$

The pitch of every tone in the whole series is, therefore, known when that of any one is known.

The major Third of \(C\), on the other hand, will be expressed by \(E_1\), that of \(F\) by \(A_1\) and so on. Hence the series of tones

$$ \begin{gathered} B\flat_1 \; + \; D_1 \; - \; F \; + \; A_1 \; - \; C \; + \; E_1 \; - \; G \; + \; B_1 \; - \; D \; + \; F_1\sharp \; - \; A, \; \&c., \end{gathered} $$

is a series of alternate major and minor Thirds. It is therefore clear that the Tones

$$ \begin{gathered} D_1 \; \pm \; A_1 \; \pm \; E_1 \; \pm \; B_1 \; \pm \; F_1\sharp , \; \&c., \end{gathered} $$

also form a series of perfect Fifths.

We have already found that the tone \(D_1\), that is the minor Third below or major Sixth above \(F\), is lower in pitch than the tone \(D\), which would be reached by a series of Fifths from \(F\), and that the difference of pitch is that known as a comma, the numerical value of which is \(\small{\tfrac{81}{80}}\), or musically about the tenth part of a whole Tone.[58] Since, then, \(D\pm A\) and \(D_1 \pm A_1\) are both perfect Fifths, \(A\) must be also a comma higher than \(A_1\) (and so also every letter with an inferior number, as 1, 2, 3, &c., attached to it, will represent a tone which is 1, 2, 3, &c., commas lower in pitch than that represented by the same letter with no inferior number attached, as is easily seen by carrying on the series.

A major triad will therefore be written thus:

$$ \begin{gathered} C + E_1 - G \end{gathered} $$

and a minor triad

$$ \begin{gathered} A_1 - C + E_1 \quad \text{or} \quad C_1 - E\flat + G_1 \end{gathered} $$

Now if we lay it down as a rule that as every inferior figure, 1, 2, 3, &c., depresses its tone by the 1, 2, 3, &c., comma, every superior figure, 1, 2, 3, &c., shall raise its tone by the same 1, 2, 3, &c., commas, we may write the major triad as

$$ \begin{gathered} c + e_1 - g \quad \text{or} \quad c^1 + e -g^1 \end{gathered} $$

and the minor triad as

$$ \begin{gathered} c - e^1\flat + g \quad \text{or} \quad c_1 - e\flat + g_1, \end{gathered} $$

or even

$$ \begin{gathered} c^1-e^2\flat-g^1 \quad \text{or} \quad c_2 - e_1\flat - g_2 \end{gathered} $$

The three series of Tones directly related to \(C\) are consequently to be written thus:

$$ \begin{gathered} C \; \text{\textemdash} \; \text{\textemdash} \; E_1 \; \text{\textemdash} \; F \; \text{\textemdash} \; G \; \text{\textemdash} \; A_1 \; \text{\textemdash} \; \text{\textemdash} \; c \end{gathered} $$ $$ \begin{gathered} C \; \text{\textemdash} \; \text{\textemdash} \; E^1\flat \; \text{\textemdash} \; F \; \text{\textemdash} \; G \; \text{\textemdash} \; A_1 \; \text{\textemdash} \; \text{\textemdash} \; c \end{gathered} $$ $$ \begin{gathered} C \; \text{\textemdash} \; \text{\textemdash} \; E^1\flat \; \text{\textemdash} \; F \; \text{\textemdash} \; G \; \text{\textemdash} \; A^1\flat \; \text{\textemdash} \; \text{\textemdash} \; c \end{gathered} $$

and the intercalary tones are —

Between the tonic and Third, \(D\), \(D_1\), or \(D^1\flat\).
Between the Sixth and Octave, \(B_1\) and \(B\flat\) or \(B^1\flat\)

Consequently the melodic tonal modes of the ancient Greeks and old Christian Church are,[60]

1) Major Mode,
$$ \begin{gathered} C...D...E_1...F...G...A_1...B_1...c \\ D_1 \hspace{3cm} \end{gathered} $$ 2) Mode of the Fourth,
$$ \begin{gathered} C...D...E_1...F...G...A_1...B\flat...c \\ \; D_1 \hspace{2.7cm} B^1\flat \end{gathered} $$ 3) Mode of the minor Seventh,
$$ \begin{gathered} C...D...E^1\flat...F...G...A_1...B\flat...c \\ \; D_1 \hspace{2.9cm} B^1\flat \end{gathered} $$ 4) Mode of the minor Third,
$$ \begin{gathered} C...D...E^1\flat...F...G...A^1\flat...B\flat...c \\ \; D_1 \hspace{3cm} B^1\flat \end{gathered} $$ 5) Mode of the minor Sixth,
$$ \begin{gathered} C...D^1\flat...E^1\flat...F...G...A^1\flat...B\flat...c \\ \hspace{3.8cm} B^1\flat \end{gathered} $$

By this notation, then, the intonation is always exactly expressed, and the kind of consonance which each tone makes with the tonic, or the tones related to it is clearly shewn.

In the ancient Greek Pythagorean intonation these scales would have to be written:

$$ \begin{gathered} \textit{Major mode}- C...D...E...F...G...A...B...C. \end{gathered} $$

and the others in a similar manner, all with letters of the same kind, belonging to the same series of Fifths.[61]

In the formulae here given for the diatonic tonal modes, the intonation of the Second and Seventh is partly undetermined. In these cases I have given \(D\) the preference over \(D_1\), and \(B\flat\) the preference over \(B^1\flat\), because the relationship of the Fifth is closer than that of the Third; but \(B\flat\) and \(D\) stand in the relation of the Fifth respectively to \(F, G\), the tones nearest related to the tonic, while \(D_1\) and \(B^1\flat\) are only in the relation of the minor Third to \(F\) and \(G\). But this reason is certainly not sufficient entirely to exclude the two last tones in homophonic vocal music. For if in a melodic phrase, the Second of the tonic came into close connection with tones related to \(F\) — for instance, if it fell between \(F\) and \(A_1\), or followed them — an accurate singer would certainly find it more natural to use the \(D_1\) which is directly related to \(F\) and \(A_1\) than the \(D\) which is related to them only in the third degree. The slightly closer relationship of the latter to the tonic could scarcely give the decision in its favour in such a case.

This ambiguity in the intercalary tones cannot, I think, be considered as a fault in the tonal system, since in our modern minor mode, the Sixth and Seventh of the tonic are often altered, not merely by a comma, but by a whole Semitone, according to the direction of the melodic progression. We shall find, however, more decisive reasons for the use of \(D\) in place of \(D_1\) in the next chapter, when we pass from homophonic music to the influence of harmonic music on the scales.

The account here given of the rational construction of scales and the corresponding intonation of intervals, deviates essentially from that given to the Greeks by Pythagoras, which has thence descended to the latest musical theories, and even now serves as the basis of our system of musical notation. Pythagoras constructed the whole diatonic scale from the following series of Fifths:

$$ \begin{gathered} F \; \pm \; C \; \pm \; G \; \pm \; D \; \pm \; A \; \pm \; E \; \pm \; B, \end{gathered} $$

and calculated the intervals from it as they have been given above. In his diatonic scale there are but two kinds of small intervals, the whole Tone \(\small{\tfrac98}\), [= 204 cents] and the Limma \(\small{\tfrac{256}{243}}\), [=90 cents].[62]

In this series if \(C\) be taken as the tonic, \(A\) would be related to the tonic in the Third degree, \(E\) in the Fourth, and \(B\) in the Fifth. Such a relationship would be absolutely insensible to any ear that has no guide but direct sensation.

A series of Fifths may certainly be tuned on any instrument, and continued as far as we please; but neither singer nor hearer could possibly discover in passing from \(c\) to \(e\) that the latter is the fourth from the former in the series of Fifths. Even in a relation of the second degree through Fifths, as of \(c\) to \(d\), it is doubtful whether a hearer can discover the relation of the tones. But in this case when we pass from one tone to the other we can imagine the insertion of 'a silent \(g\),' so to speak, forming the Fourth below \(c\), and the Fifth below \(d\), and thus establish a connection, for the mind's ear at least, if not for the body's. This is probably the meaning to be attached to Rameau's and d'Alembert's explanation that a singer effects the passage from \(c\) to \(d\) by means of the fundamental bass \(G\). If the singer does not hear the bass note \(G\) at the same time as \(d\), he cannot possibly bring his \(d\) into consonance with that \(G\); but the melodic progression may certainly be facilitated by conceiving the existence of such a tone. This is a well-known means for striking the more difficult intervals, and is often applied with advantage. But of course it completely fails when the transition has to be made between tones widely separated in the series of Fifths.[63]

Finally there is no perceptible reason in the series of Fifths why they should not be carried further, after the gaps in the diatonic scale have been supplied. Why do we not go on till we reach the chromatic scale of Semitones? To what purpose do we conclude our diatonic scale with the following singularly unequal arrangement of intervals —

$$ 1, \; 1, \; \tfrac12 \;, 1, \; 1, \; 1, \; \tfrac12 $$

The new tones introduced by continuing the series of Fifths would lead to no closer intervals than those which already exist. The old scale of five tones appears to have avoided Semitones as being too close. But when two such intervals already appear in the scale, why not introduce more?

The Arabic and Persian musical system, so far as its nature is shewn in the writings of the older theorists, also knew no method of tuning but by Fifths. But this system, which seems to have developed its peculiarities in the Persian dynasty of the Sassanides (A.D. 226-651) before the Arabian conquest, shews an essential advance on the Pythagorean system of Fifths.

In order to judge of this system of music, which has been hitherto completely misunderstood, the following relation has to be known. By tuning four Fifths upwards from \(C\)

$$ \begin{gathered} C \; \pm \; G \; \pm \; D \; \pm \; A \; \pm \; E \end{gathered} $$

we come to a tone, \(E\), which is \(\tfrac{81}{80}\) or a comma higher than the natural major Third of \(C\), which we write \(E_1\). The former \(E\) forms the major Third in the Pythagorean scale. But if we tune eight Fifths downwards from \(C\), thus —

$$ \begin{gathered} C \; \pm \; F \; \pm \; B\flat \; \pm \; E\flat \; \pm \; A\flat \; \pm \; D\flat \; \pm \; G\flat \; \pm \; C\flat \; \pm \; F\flat \end{gathered} $$

we come to a tone, \(F\flat\), which is almost exactly the same as the natural \(E_1\). The interval of \(C\) to \(F\flat\) is expressed by

$$ \small{\tfrac{8192}{6561}=\tfrac54 \times \tfrac{32768}{32805}}, \quad \text{or nearly} \quad \small{\tfrac54 \times \tfrac{885.6}{886.6}} \quad \text{[=384 cents].} $$

Hence the tone \(F\flat\) is lower than the natural major Third \(E_1\) [= 386 cents] by the extremely small interval \(\small{\tfrac{887} {886}}\) [= 2 cents], which is about the eleventh part of a comma [ = 22 cents]. This interval between \(F\flat\) and \(E^1\) is practically scarcely perceptible, or at most only perceptible by the extremely slow beats produced by the chord \(C...F\flat...G\) [ \(=C \;384\; F\flat\; 318\; G\)] upon an instrument most exactly tuned. Practically, then, we may without hesitation assume that the two tones \(F\flat\) and \(E_1\) are identical, and of course that their Fifths are also identical, or

$$ F\flat=E_1, \; C\flat=B_1, \; G\flat=F_1\sharp, \; \&c. $$

Now in the Arabic and Persian scale the Octave is divided into 16 intervals, but in our equal temperament it is divided into 6 whole Tones. Modern [European] interpreters of the Arabic and Persian system of music have hence been misled into the conclusion that each of the 17 degrees of the scale corresponded to about the third of a Tone in our music. In that case the intonation of the degrees in the Arabic and Persian scale would not be executable on our instruments. But in Kiesewetter’s work on the music of the Arabs,[65] which was written with the assistance of the celebrated Orientalist von Hammer-Purgstall, there is given a translation of the directions for the division of the monochord laid down by Abdul Kadir, a celebrated Persian theorist of the fourteenth century of our era, that lived at the courts of Timur and Bajazet. These directions enable us to calculate the intonation of the Oriental scale with perfect certainty. These directions also agree in essentials both with those of the much older Farabi,[66] (who died in A.D. 950), and of his own contemporary, Mahmud Shirazi,[67] (who died in 1315), for dividing the fingerboard of lutes. According to the directions of Abdul Kadir all the tonal degrees of the Arabic scale are obtained by a series of 16 Fifths, and if we call the lowest degree \(C\), and arrange them in order of pitch within the compass of an Octave, they will be the following, as expressed in our notation [with the addition of the grave accent explained in p. 281, note footnote 63].

$$ \begin{gathered} 1) \;C\text{\textemdash} \;2) \;D\flat\text{\textemdash}\;3) \; ^{\backprime}D_1\smile \; 4) \;D\text{\textemdash}\; 5) \;E\flat\text{\textemdash} \; 6) \; ^{\backprime}E_1 \smile \\ 7) \;E\text{\textemdash} \;8)\; F\text{\textemdash} \; 9)\; G\flat \text{\textemdash} \; 10)\; ^{\backprime}G_1 \smile \; 11)\; G \text{\textemdash} \; 12)\; A\flat \text{\textemdash} \\ 13)\; ^{\backprime}A_1 \smile \; 14)\; A \text{\textemdash} \; 15)\; B\flat \text{\textemdash} \; 16)\; ^{\backprime}B_1 \text{\textemdash} \; 17)\; ^{\backprime}c_1 \smile \; 18)\; c \end{gathered} $$

where the line - between two tones indicates the interval of a Pythagorean Limma \(\small{\tfrac{256}{243}}\)(which is nearly \(\small{\tfrac{20}{19}}\) [= 90 cents]), and the sign \(\smile\) a Pythagorean comma [= 531441 : 524288 = 24 cents]. The Limma is about \(\small{\tfrac45}\) and the Pythagorean comma a little more than \(\small{\tfrac15}\) and less than \(\small{\tfrac29}\) of the natural Semitone \(\small{\tfrac{16}{15}}\) [= 112 cents].

Abdul Kadir assigns the following intonation to the three first of the 12 principal tonal modes or Makamat: —

Arabic   Ancient Greek
1. Uschak \(C...D...E...F...G...A...B\flat ...c\) Hypophrygian or Ionic.
2. Newa \(C...D...E\flat ...F...G...A\flat ...B\flat ...c\) Hypodorian or Eolic.
3. Buselik \(C...D\flat ...E\flat ...F...G\flat ...A\flat ...B\flat ...c\) Mixolydian [all on p. 269].

These three are therefore completely identical with the ancient Greek scales in Pythagorean intonation.[68] Since the Arabic theoreticians divide these scales into the Fourth \(C...F\) and the Fifth \(F \pm c\), and since \(C, \; F\) and \(B\flat\) are considered to be invariable tones, and the others to be variable, it is probable that \(F\) must be regarded as the tonic. In this case

          1. Uschak would be = \(F\) major.

          2. Newa would be = the mode of the minor Seventh of \(F\). [69]

          3. Buselik would be = the mode of the minor Sixth of \(F\).

all three in Pythagorean intonation. The Persian school also considers the scales to be related.

The next group consists of five tonal modes having just or natural intonation, namely:

4. Rast \(C ... ^{\backprime}D_1 ... ^{\backprime}E_1 ... F ... G ...^{\backprime}A_1 ... B\flat ... c\)
5. Husseini \(C ... ^{\backprime}D_1 ... E\flat ... F ... ^{\backprime}G_1 ... A\flat ... B\flat ... c\)
6. Hidschaf \(C ... ^{\backprime}D_1 ... E\flat ... F ... ^{\backprime}G_1 ... ^{\backprime}A_1 ... B\flat ... c\)
7. Rahewi \(C ... ^{\backprime}D_1 ... ^{\backprime}E_1 ... F ... ^{\backprime}G_1 ... A\flat ... B\flat ... c\)
8. Sengule \(C ... D ... ^{\backprime}E_1 ... F ... ^{\backprime}G_1 ... ^{\backprime}A_1 ... B\flat ... c\)

Rast may be regarded as the mode of the Fourth of \(C\); Hidschaf as the mode of the Fourth of \(F\); Husseini as the mode of the Fourth of \(B\flat\); as such they would have perfectly natural intonation. In Rahewi, if we refer it to the tonic \(F\), the minor Third \(A\flat\) is in Pythagorean, not natural, intonation. It might be regarded as the mode of the minor Seventh of \(F\) in which the major Seventh \(E_1\) is used as the leading note in place of the minor Seventh, as in our own minor mode. The natural intonation of such a tonal mode cannot, indeed, be properly represented by the existing 17 tonal degrees. It becomes necessary to take either Pythagorean minor Thirds and natural major Thirds or conversely. Husseini may be regarded as the same tonal mode with Rahewi, having the same false minor Third, but a minor Seventh. Finally Sengule may be regarded as \(F\) major with a Pythagorean Sixth. Rast may be conceived in the same way; they are merely distinguished by the different values of the Seconds \(G\) or \(^{\backprime}G_1\)

The four last Makamat have each 8 tones, new intercalary tones being employed. Two of them resemble the modes Rast and Sengule, and between \(B\flat\) and \(C\) there is an intercalary tone \(^{\backprime}c_1\), introduced; named

9. Irak \(C ... ^{\backprime}D_1 ... ^{\backprime}E_1 ... F ... G ...^{\backprime}A_1 ... B\flat ... ^{\backprime}c_1 ... c\)
10. Iszfahan \(C ... D ... ^{\backprime}E_1 ... F ... ^{\backprime}G_1 ... ^{\backprime}A_1 ... B\flat ... ^{\backprime}c_1 ... c\)
The last transposed a Fourth gives
11. Büsürg \(C ... D ... ^{\backprime}E_1 ... F ... ^{\backprime}G_1 ... G ... A ... ^{\backprime}B_1 ... c\)
The last tonal mode is
12. Zirefkend \(C ... ^{\backprime}D_1 ... E\flat ... F ... ^{\backprime}G_1 ... A\flat ... ^{\backprime}A_1 ... ^{\backprime}B_1 ... c\)

which certainly, if rightly reported, is a very singular creation. It might be looked upon as a minor scale with a major Seventh, and both a major and minor Sixth, but then the Fifth \(^{\backprime}G_1\) is wrong. On the other hand, if \(F\) is taken as the tonic, it has no Fourth, for which certainly there is some analogy in the Mixolydian and Hypolydian scales. The instructions for scales of eight notes are very contradictory, to judge by the different authorities cited by Kiesewetter.

The following four are distinguished as the principal modes of the Makamat: —

     1. Uschak = Pythagorean \(F\) major.

     2. Rast = Natural mode of the Fourth of C, or natural F major with acute Sixth.

     3. Husseini = Natural mode of the minor Seventh of \(F\)

     4. Hidschaf = Natural mode of the Fourth of \(F\)

Keyboard for Persian Modes

We find, then, a decided predominance of scales with a perfectly correct natural intonation, which has been attained by a skilful use of a continued series of Fifths. This makes the Arabic and Persian tonal system very noteworthy in the history of the development of music. Moreover, in some of these scales we find ascending leading notes, which are perfectly foreign to the Greek scales. Thus in Rahewi, \(E_1\) is the leading note to \(F\), although the minor Third \(A\flat\) stands above \(F\), while no Greek scale could have allowed this without at the same time changing \(E_1\) into \(E\flat\). Similarly in Zirefkend the \(B_1\), is used as a leading note to \(C\), although the minor Third \(E\flat\) is used above \(C\). [70]

At a little later period a new musical system was developed in Persia with 12 Semitones to the Octave, analogous to the modern European system. Kiesewetter here hazards the very unlikely hypothesis that this scale was introduced into Persia by Christian missionaries. But it is clear that the system of 17 tonal degrees which had been previously in popular use, merely required the feeling for the finer distinctions to grow dull so that intervals which differed only by a [Pythagorean] comma should be confused, in order to generate the system of 12 Semitones.[71] No foreign influence was necessary here. Moreover, the Greek system of music had long been taught to the Arabs and Persians by Alfārābī. Again, the European theory of music had not made any essential advance in the fourteenth and fifteenth centuries, if we except the study of harmony, which never found favour with the Orientals. Hence the Europeans of those days could teach the Orientals nothing that they did not already know better themselves, except some imperfect rudiments of harmony which they did not want. There is much more reason, I think, for asking whether the imperfect fragments of the natural system which we find among the Alexandrine Greeks, do not depend on Persian traditions, and also, whether the Europeans in the time of the Crusades did not learn much music from the Orientals. It is very probable that they brought the lute-shaped instruments with fingerboards and the bowed instruments from the East. In the construction of tonal modes we might especially instance the use of the leading note, which we have here found existing in the East, and which at that time also began to figure in the Western music.

The use of the major Seventh of the scale as a leading note to the tonic marks a new conception, which admitted of being used for the further development of the tonal degrees of a scale, even within the domain of purely homophonic music. The tone \(B_1\) in the major scale of \(C\) has the most distant relationship of all the tones to the tonic \(C\), because as the major Third of the dominant \(G\), it has a less close connection with it than its Fifth \(D\). We may perhaps assume this to be the reason why, when a sixth tone was introduced into some Gaelic airs, the Seventh was usually omitted. But, on the other hand, the major Seventh \(B_1\) developed a peculiar relation to the tonic, which in modern music is indicated by calling it the leading note. The major Seventh \(B_1\) differs from the Octave \(c\) of the tonic by the smallest interval in the scale, namely a Semitone, and this proximity to the tonic allows the Seventh to be struck easily and pretty surely, even when starting from tones in the scale which are not at all related to \(B_1\). The leap \(F...B_1\) [= 45 : 32 = 590 cents], for example, is difficult, because there is no relationship at all between the tones. But when a singer has to perform the passage \(F...B_1...c\), he conceives the interval \(F...c\) , which he can easily execute, but does not force his voice up sufficiently high to reach \(c\) at first, and thus strikes \(B_1\) on the way. Thus \(B_1\) assumes the appearance of a preparation for \(c\), and this view alone justifies it to the ears of a listener, by whom the transition into \(c\) is, therefore, expected. Hence it has been said that \(B_1\) leads to \(c\); or that \(B_1\) is the leading note to \(c\). In this sense it also becomes easy to sharpen \(B_1\) somewhat, making it \(B\), for example, to bring it near to \(c\), and mark its reference to that tone more distinctly.

According to my own feeling, the leading effect of the tone \(B_1\) is much more marked in such passages as \(F...B_1..c\) or \(F + A_1...B_1...c\), in which \(B_1\) is not related to the preceding tones, than in such a passage as \(G + B_1...c\) where it is. But as I have found nothing on this point in musical writings, I do not know whether musicians are likely to agree with me in this opinion. For the other Semitone of the scale \(E_1...F\), the \(E_1\) does not seem to lead to \(F\), if the tonality of the melody is well preserved, because in this case \(E_l\) has its own independent relation to the tonic, and hence is musically quite determinate. The hearer, then, has no occasion to regard \(E_1\) as a mere preparation for \(F\). Similarly for the interval \(G...A^1\flat\) [ = 112 cents] in the minor mode. The \(G\) is more nearly related to the tonic \(C\) than \(A^1\flat\) is. On the other hand, Hauptmann is probably right in considering the interval \(D...E^1\flat\) [= 112 cents] in the minor mode, as one in which \(D\) leads to \(E^1\flat\), because \(D\) has only a relationship of the second degree to the tonic \(C\), although its relationship is certainly closer than that of \(B_1\) to \(C\).

But the relation of \(D^1\flat\) in descending passages of the mode of the minor Sixth of \(C\) (the old Greek Doric) is perfectly similar to the effect of \(B_1\) in the ascending scale of \(C\) major. It really forms a kind of descending leading note, and since in the best period of Greek music descending passages were felt as nobler and more harmonious than ascending ones,[72] this peculiarity of the Doric mode may have been of special importance and have been a reason for the preference given to this scale. The cadence with the chord of the extreme sharp Sixth [ratio 128 : 225, cents 976] —

$$ \begin{gathered} D^1\flat + F \quad ... G + B_1 \\ C \; - E^1\flat + G ... c \end{gathered} $$

is almost the only remnant of the ancient tonal modes. It is quite isolated and misunderstood. This is a (Greek) Doric cadence, in which \(D^1\flat\) and \(B_1\) are both used at the same time as leading notes to \(C\).[73]

The relation of the second or parhypatē of the Greek Doric scale, to the lowest tone or hypatē, seems also to have been perfectly well felt by the Greeks themselves, to judge by Aristotle's remarks in the 3rd and 4th of his problems on harmony. I cannot abstain from adducing them here because they admirably and delicately characterise the relation. Aristotle inquires why the singer feels his voice more taxed in taking the parhypatē than in taking the hypatē, although they are separated by so small an interval. The hypatē is sung, he says, with a remission of effort. And then he adds that in order to reach an aim easily it is necessary that in addition to the motive which determines the will, the kind of volitional effort should be quite familiar and easy to the mind.[74] The effort felt in singing the leading note does not lie in the larynx, but in the difficulty we feel in fixing the voice upon it by mere volition while another tone is already in our mind, to which we desire to pass, and which by its proximity conducted us to the leading note. It is not till we reach the final tone that we feel ourselves at home and at rest, and this final tone is sung without any strain on the will.

Proximity in the scale then gives a new point of connection between two tones, which is not merely active in the case of the leading note, just considered, but also, as already mentioned, in interpolating tones between two others in the chromatic and enharmonic modes. Intervals of pitch are in this respect analogous to measurements of distance. When we have the means of determining one point (the tonic) with great exactness and certainty, we are able by its means to determine other points with certainty, when they are at a known small distance from it (the interval of a Semitone), although perhaps we could not have determined them with so much certainty independently. Thus the astronomer employs his fundamental fixed stars, of which the positions have been determined with the greatest possible accuracy, for accurately determining the positions of other stars in their neighbourhood.

We may also remark that the interval of a Semitone plays a peculiar part as the introduction (appoggiatura) to another note. As an appoggiatura in a melody any tone can be used, even when not in its scale, provided it makes the interval of a Semitone with a note in the scale which it introduces; but a foreign tone which makes the interval of a whole Tone with that note in the scale, cannot be so used. The only justification of this use of the Semitone is certainly its existence as a well-known interval in the diatonic scale, which the voice can sing correctly and the ear can readily appreciate, even when the relations on which its magnitude depends are not clearly sensible in the passage where it is used. Hence also no arbitrarily chosen small interval can be thus employed. Although slight changes in the interval of the leading note may be introduced by practical musicians to give a stronger expression to its tendency towards the tonic, they must never go so far as to make those changes clearly felt.[75]

Hence the major Seventh in its character of leading note to the tonic acquires a new and closer relationship to it, unattainable by the minor Seventh. And in this way the note which is most distantly related to the tonic becomes peculiarly valuable in the scale. This circumstance has continually grown in importance in modern music, which aims at referring every tone to the tonic in the clearest possible manner; and hence, in ascending passages going to the tonic, a preference has been given to the major Seventh in all modern keys, even in those to which it did not properly belong. This transformation appears to have begun in Europe during the period of polyphonic music, but not in part songs only, for we find it also in the homophonic Cantus firmus of the Roman Catholic Church. It was blamed in an edict of Pope John XXII., in 1322, and in consequence the sharpening of the leading note was omitted in writing, but was supplied by the singers, a practice which Winterfeld believes to have been followed by Protestant musical composers even down to the sixteenth and seventeenth centuries, because it had once come into use. And this makes it impossible to determine exactly what were the steps by which this change in the old tonal modes was effected.[76]

Even to the present day, according to A. v. Oettingen’s report,[77] the Esthonians struggle against singing the leading note in minor scales, although it may be clearly struck on the organ.

Among the ancient tonal modes, the Greek Lydian (major mode) and the unmelodic Hypolydian (mode of the Fifth, p. 269, No. 7) had the major Seventh as the leading note to the tonic, and hence the first was developed into the principal tonal mode of modern music, the major mode. The Greek Ionic (mode of the Fourth) differed from it only in having a minor Seventh. On simply altering this into a major Seventh, this mode also became major. On giving a major Seventh to each of the other three, they gradually converged to our present minor mode during the seventeenth century. From the Greek Phrygian (mode of the minor Seventh) by changing \(B\flat\) into \(B_1\) we obtain

$$ \begin{gathered} C...D...E^1\flat ...F...G...A_1...B_1...c \end{gathered} $$

as we had already found from a simple consideration of the relationship of tones [p. 274, No. 2], The Greek Hypodoric or Eolic (mode of the minor Third), which answers to our descending minor scale, gives on changing \(B^1\flat\) into \(B_1\)

$$ \begin{gathered} C...D...E^1\flat ...F...G...A^1\flat ...B_1...c \end{gathered} $$

which is difficult for singers to execute, on account of the interval \(A^1\flat ... B_1\)[= ratio 64 : 75, cents 274], but frequently occurs in modern music both ascending and descending.

The Greek Doric (mode of the minor Sixth) with a major instead of a minor Seventh, is still discoverable in the final cadence mentioned on p. 286.

The general introduction of the leading tone represents, therefore, a continually increasing consistency in the development of a feeling for the predominance of the tonic in a scale. By this change, not only is the variety of character in the ancient tonal modes seriously injured, and the wealth of previous means of expression essentially diminished, but even the links of the chain of tones in the scale were disrupted or disturbed. We have seen that the most ancient theory made tonal system consist of series of Fifths, and that each system had at first four and afterwards six intervals of a Fifth. The predominance of a tonic as the single focus of the system was not yet indicated, at least externally; it became apparent at most by a limitation of the number of Fifths to contain those tones only which occurred in the natural scale. All Greek tonal modes may be formed from the tones in the series of Fifths —

$$ \begin{gathered} F \; \pm \; C \; \pm \; G \; \pm \; D \; \pm \; A \; \pm \; E \; \pm \; B \end{gathered} $$

Directly we proceed to the natural intonation of Thirds, the series of Fifths is interrupted by an imperfect Fifth, as in

$$ \begin{gathered} F \; \pm \; C \; \pm \; G \; \pm \; D \; ... \; A_1 \; \pm \; E_1 \; \pm \; B_1 \end{gathered} $$

where the Fifth \(D...A_1\) [= 680 cents] is imperfect. And when finally the sharp leading note is introduced, as by the use of \(G_2\sharp\) for \(G\) in \(A_1\) minor, the series is entirely interrupted [\(C : G_2\sharp\) = 16 : 25 = 772 cents] .

In the gradual development of the diatonic system, therefore, the various links of the chain which bound the tones together were sacrificed successively to the desire of connecting all the tones in a scale with one central tone, the tonic. And in exact proportion to the degree with which this was carried out, the conception of tonality consciously developed itself in the minds of musicians.

The further development of the European tonal system is due to the cultivation of harmony, which will occupy us in the next chapter.

But before leaving our present subject, some doubtful points have still to be considered. In the preceding chapter I have shewn that the melodic relationship of tones can be made to depend upon their upper partials, precisely in the same way as their consonance was shewn to be determined in Chapter X. Now this method of explanation may in a certain sense be considered to agree with the favourite assertion that 'melody is resolved harmony,' on which musicians do not hesitate to form musical systems without staying to inquire how harmonies could have been resolved into melodies at times and places where harmonies had either never been heard, or were, after hearing, repudiated. According to our explanation, at least, the same physical peculiarities in the composition of musical tones, which determined consonances for tones struck simultaneously, would also determine melodic relations for tones struck in succession. The former then would not be the reason for the latter, as the above phrase suggests, but both would have a common cause in the natural formation of musical tones.

Again, in consonance we found other peculiar relations, due to combinational tones, which become effective even when simple tones, or tones with few and faint upper partials, are struck simultaneously. I have already shewn that combinational tones very imperfectly replace the effect of upper partial tones in a consonance, and that consequently a chord formed of simple tones is wanting in brightness and character, the distinctions between consonance and dissonance being only very imperfectly developed.

In melodic passages, however, combinational tones do not occur, and hence the question arises as to how far a melodic effect could be produced by a succession of simple tones. There is no doubt that we can recognise melodies which we have already heard, when they are executed on the stopped pipes of an organ, or are whistled with the mouth, or merely struck on a glass or wood or steel harmonicon, as a musical box, or are played on bells. But there is also no doubt that all these instruments, which generate simple tones, either alone or accompanied by weak and remote inharmonic secondary tones, are incapable of producing any effective melodic impression without an accompaniment of musical instruments proper. They may be often extremely effective for performing single parts when accompanied by the organ, or the orchestra, or a pianoforte, but by themselves they produce very poor music indeed, which degenerates into absolute unpleasantness when the inharmonic secondary tones are somewhat too loud.

We are bound, however, to give some reason why any impression of melody at all can be produced by such instruments.

Now we must first remember that, as shewn at the end of Chapter VII., the actual construction of the ear favours the generation of weak harmonic upper partials within the ear itself, when powerful but objectively simple tones are sounded. Hence it is at most very weak objectively simple tones which can be regarded as also subjectively simple.

Next, there is an effect of memory to be brought into account. Supposing that I have been used to hear Fifths taken at all possible pitches, and have recognised them by aural sensation as having a very close melodic relationship, I should know the magnitude of this interval by experience for every tone in the scale, and should retain the knowledge thus acquired by the action of a man's memory of sensations, even of those for which he has no verbal expression.

When, then, I hear such an interval executed on tuning-forks, I am able to recognise it as an interval I have often heard, although its tones have either none, or only some faint remnants of those upper partials which formerly gave it a right to be considered as a favourite interval of close melodic relationship. And just in the same way I shall be able to recognise, as previously known, other melodic passages or whole melodies which are executed in simple tones, and even if I hear a melody for the first time in this way, whistled with the mouth or chimed by a clock, or struck on a glass harmonicon, I should be able to complete it by imagining how it would sound if executed on a real musical instrument, as the voice or a violin. A practised musician is able to form a conception of a melody by merely reading the notes. If we give the prime tones of these notes on a glass harmonicon, we give a firmer basis to the conception by really exciting a large portion of the impression on the senses which the melody would have produced if sung. Simple tones, however, merely exhibit an outline of the melody. All that gives the melody its charm is absent. We know, indeed, the individual intervals which it contains, but we have no immediate impression on our senses which serves to distinguish those which are distantly from those which are closely related, or the related from the totally unrelated. Observe the great difference between merely whistling a melody or playing it on a violin; between striking it on a glass harmonicon or on the piano! The difference is somewhat of the same kind as that between viewing a single photograph of a landscape, and seeing two corresponding photographs of it through a stereoscope. The first enables me, by means of my memory, to form a conception of the relative distances of its parts, and this conception may be often very satisfactory. But the stereoscopic fusion of the two figures gives me the real impression on the senses which the relative distances of the parts of the landscape would have themselves produced, and which I am obliged in the case of a single image to supply by experience and memory. Hence the stereoscopic picture is more lively than the simple perspective view, exactly in the same way as immediate impressions on our senses are more lively than our recollections.

The case seems to be the same for melodies executed in simple tones. We recognise the melodies when we have heard them otherwise performed; we can even, if we have sufficient musical imagination, picture to ourselves how they would sound if executed by other instruments, but they are decidedly without the immediate impression on the senses which gives music its charm.

[1]Hanslick seems to me to have the advantage over other esthetic writers in this point, because music, unassisted by poetry, has no means of clearly characterising the object of feeling.
[2][The above words conclude the problem, which it seems best to cite in full. Which we may perhaps translate thus: 'Why is sound the only sensation which excites the feelings? Even melody without words has feeling. But this is not the case for colour, or smell, or taste. Is it because they have none of the motion which sound excites in us? For the others excite motion; thus colour moves the eye. But we feel the motion which follows sound. And this is alike, in rhythm, and alteration in pitch, but not in united sounds. Sounding notes together does not excite feeling. This is not the case for other sensations. Now these motions stimulate action, and this action is the sign of feeling.' Aristotle seems to have required motion to excite feeling, and in sounding two notes together, there was no motion of one towards the other. It is evident that he had not the slightest inkling of a progression of harmonies, and this utter blank in his mind is one of the strongest proofs that the Greeks had never tried harmony. 'Αρμονία (pronounced 'harmonia') had the modern meaning of melody; μελωδία (pronounced 'melodia') was words set to music. — Translator.]
[3] [It will be seen in App. XX. sect. K. that the Fourth and Fifth are often materially inexact or designedly altered. — Translator.]
[4][Some considerations have been omitted, probably by design. The quality of tone of the voice which sings the Octave above is materially different. The evenly numbered partials of the lower tone are by no means so powerful as in the higher tone. The upper partials of the higher tone, which are still quite effective, would be inaudible in the lower tone. — Translator.]
[5][This applies to the Pythagorean scale and hence to Greek music, and also to all tempered music. But in just intonation \(c\) to \(d\) is a major Tone, and \(d\) to \(e\) a minor Tone, whereas \(g\) to \(a\) is a minor Tone and \(a\) to \(b\) a major Tone. These distinctions were of course purposely omitted in the text. — Translator.]
[6][But see App. XX. sect. K. — Translator.]
[7][Other experimental instruments will be described in App. XX. sect. F. The Harmonical gives only the just major Third 4 : 5. Its nearest approach to the Pythagorean 64 : 81, or 408 cents, is \(B\): \(D_1\) = 68 : 80, or 413 cents, not so harsh but quite near enough to shew its character. For the intervals used by violinists, see also App. XX. sect. G. arts. 6 and 7. — Translator.]
[8]Histoire Générale de la Musique, Paris, 1869, vol. i.
[9] [See App. XX. sect. K. for pentatonic scales in Java, China, and Japan. — Translator.]
[10]Descriptions des Instruments de Musique des Orientaux; chap. xiii. in the Description de l'Egypte. État Moderne.
[11][This is probably only a rude approximation or a guess. See App. XX. sect. K. for observations on existing pentatonio scales actually heard. — Translator.]
[12] Nicomachus makes Philolaus say (edit. Meibomii, p. 17), 'From the Hypatē (e) to the Mesē (a) was a Fourth, from the Mesē (a) to the Nētē (e') a Fifth, from the Nētē (e') to the Tritē (b) a Fourth, from the Tritē (b) to the Hypatē (e) a Fifth'. This shews that c, not b, was the missing note.
[13][The upper tetrachord was thus reduced to a trichord, while the lower remained a perfect tetrachord. If we take Pythagorean into nation the cents are \(e\;90\; f\; 204\; g\; 204\; a\; 204\; b\; 294\; d'\; 204\; e'\). — Translator.]
[14] [Taking Pythagorean intonation, the cents in the intervals are \(b\; 90\; c \;408\; e \;90\; f\; 408\; a \;204\; b\). The account of the popular tuning of the Ko-to, the national Japanese instrument, furnished by the Japanese, but in European notes, at the International Health Exhibition in London, 1884, gives many varieties of this scale, see App. XX. sect. K. Japan. — Translator.]
[15]Chinese Melodies, in Ambrosch’s Geschichte der Musik, vol. i. pp. 30, 34, 35. Of Scotch melodies there is a fine collection with reference to the authorities and the older forms in G. F. Graham’s Songs of Scotland, 3 vols. Edinburgh, 1859. The modern pianoforte accompaniment which has been added, is often ill enough suited to the character of the airs.
[16][Exclusive of the two drones there are only 9 tones on the bagpipe. For the whole of these observations see App. XX. sect. K. — Translator.]
[17][In the following investigation the Author all along assumes harmonic forms of the intervals, which are certainly modern. The cents in the five forms cited, as determined from the ratios given, are: $$ \begin{alignedat}{10} &1) \hspace{3mm}&c \hspace{3mm}&204 \hspace{3mm}&d \hspace{3mm}&294 \hspace{3mm}&f \hspace{3mm}&204 \hspace{3mm}&g \hspace{3mm}&204 \hspace{3mm}&a \hspace{3mm}&316 \hspace{3mm}&c'. \\ &2) \hspace{3mm}&C \hspace{3mm}&316 \hspace{3mm}&E^1\flat \hspace{3mm}&182 \hspace{3mm}&F \hspace{3mm}&204 \hspace{3mm}&G \hspace{3mm}&294 \hspace{3mm}&B\flat \hspace{3mm}&204 \hspace{3mm}&c. \\ &3) \hspace{3mm}&c \hspace{3mm}&204 \hspace{3mm}&d \hspace{3mm}&294 \hspace{3mm}&f \hspace{3mm}&204 \hspace{3mm}&g \hspace{3mm}&294 \hspace{3mm}&b\flat \hspace{3mm}&204 \hspace{3mm}&c'. \\ &4) \hspace{3mm}&c \hspace{3mm}&204 \hspace{3mm}&d \hspace{3mm}&182 \hspace{3mm}&e_1 \hspace{3mm}&316 \hspace{3mm}&g \hspace{3mm}&182 \hspace{3mm}&a_1 \hspace{3mm}&316 \hspace{3mm}&c'. \\ &5) \hspace{3mm}&C \hspace{3mm}&316 \hspace{3mm}&E^1\flat \hspace{3mm}&182 \hspace{3mm}&F \hspace{3mm}&316 \hspace{3mm}&A^1\flat \hspace{3mm}&204 \hspace{3mm}&B^1\flat \hspace{3mm}&182 \hspace{3mm}&c. \\ &6) \hspace{3mm}&c \hspace{3mm}&316 \hspace{3mm}&e^1\flat \hspace{3mm}&182 \hspace{3mm}&f \hspace{3mm}&204 \hspace{3mm}&g \hspace{3mm}&182 \hspace{3mm}&a_1 \hspace{3mm}&316 \hspace{3mm}&c. \\ \end{alignedat} $$ Translator.]
[18]In the following way — the numbers referring to the schemes in this page, and the corresponding cents, of course, belonging to equal temperament: $$ \begin{alignedat}{30} &1) && \; &&c\sharp 200 \; &&d\sharp 300 \; &&f\sharp 200 \; &&g\sharp 200 \; &&a\sharp 300 \; &&c'\sharp \; && \; && \; &&\\ &2) && && &&d\sharp 300 &&f\sharp 200 &&g\sharp 200 &&a\sharp 300 &&c'\sharp 200 &&d'\sharp && && \\ &3) && && && && &&g\sharp 200 &&a\sharp 300 &&c'\sharp 200 &&d'\sharp 300 &&f'\sharp 200 &&g'\sharp \\ &4) && && && &&f\sharp 200 &&g\sharp 200 &&a\sharp 300 &&c'\sharp 200 &&d'\sharp 300 &&f'\sharp &&\\ &5) &&A\sharp 300 &&c\sharp 200 &&d\sharp 300 &&f\sharp 200 &&g\sharp 200 &&a\sharp && && && && \\ \end{alignedat} $$ And this shews that all five are formed by a simple succession of tempered Fifths, for the five black notes arranged in order of Fifths are \(f\sharp \hspace{2mm} 700 \hspace{2mm} c\sharp \hspace{2mm} 700 \hspace{2mm} g\sharp \hspace{2mm} 700 \hspace{2mm} d\sharp \hspace{2mm} 700 \hspace{2mm} a\sharp\). The piano being tuned in equal temperament gives very nearly perfect Fifths, and hence very well imitates a succession of five notes thus tuned. If, however, the Fifths are perfect, then every 200 and 300 cents in the above scheme becomes 204 and 294, differences which few ears will perceive in melody.
[19][Mr. Colin Brown, Euing Lecturer on the Science, Theory, and History of Music, Anderson’s College, Glasgow, in The Thistle, 'a miscellany of Scottish song, with notes critical and historical; the melodies arranged in their natural modes; with an introduction, explaining the construction and characteristics of Scottish music, the Principles, Laws, and Origin of Melody' (Glasgow, Dec. 1883), says, p. viii.: 'The pentatonic form of the scale is used in Scotland, but not to a greater extent than in the national music of some other countries. A general idea seems to prevail that Scottish music can be played upon the five black digitals of the pianoforte, representing what is popularly known as the Caledonian scale, but any one who will take the trouble to examine Scottish music will find that not more than a twentieth part of our old melodies are pentatonic, or constructed upon this form of the scale. In Dauney’s work, where the Skene MSS. (the oldest collection extant) are noted, this statement is fully verified.' I have examined the first 36 airs as printed in The Thistle, and I found only one which was strictly pentatonic, p. 51, No. 8, Lament for Ruaridh Mor, Macleod of Macleod — Dunvegan 1626. But in nearly a quarter of the airs the Semitones were introduced by an unaccented note which looked to be modern, as in Roy's Wife, p. 10, and the Banks and Braes o' Bonnie Doon, p. 48, on the last of which Mr. Brown observes, p. 49: 'With pentatonic theorists Ye Banks and Braes is a favourite example of this assumed peculiarity of Scottish music. But it can only be brought into the pentatonic scale by being played in an incomplete form.' The only places in which the Seventh \(g\sharp\) occurs are the cadence \(e' \space f'\sharp \space g'\sharp \space a'\) (which occurs twice, and is evidently out of character, and should be \(e' \space f'\sharp \space a' \space a'\)), and the flourished ad libitum cadence \(f''\sharp \space e'' \space d'' \space c''\sharp \space b''\) containing the Fourth \(d\) (which should clearly be \(f''\sharp \space e'' \space c''\sharp \space b'\)). And many of the others can be probably 'restored' in a similar fashion. Thus of Roy's Wife Mr. Brown himself says, p. 11, 'played as a dance tune it is pentatonic,' and gives the substitutes for his version, which are clearly the more ancient forms. Mr. Brown gives as the marks of Scotch music (pp. ix., x.) 1. its modal character, being constructed on the ancient seven modes; 2. its modulation or change of mode, which is constant; 3. almost absence of transition or change of key; 4. preponderance of minor forms of the scale; 5. almost absence of sharp Sevenths in the minors; 6. cadences on to every note of the scale, and double cadences closing on an unaccented note, which are simple (repeating the cadential tone) or compound (the unaccented tone differing from the preceding). — Translator.]
[20][Scale, tempered \(d \hspace{2mm} 200 \hspace{2mm} e \hspace{2mm} 300 \hspace{2mm} g \hspace{2mm} 200 \hspace{2mm} a \hspace{2mm} 200 \hspace{2mm} b \hspace{2mm} 300 \hspace{2mm} d'\) . That is, no \(f\sharp\) and no \(c\sharp\). All these scales are merely the best representatives in European notation of the sensations produced by the scales on European listeners. They cannot be received as correct representations of the notes actually played. — Translator.]
[21]Playford's Dancing Master, ed. 1721. The first edition appeared in 1657. —Songs of Scotland, vol. iii. p. 170. [Scale, \(d \hspace{2mm} 300 \hspace{2mm} f \hspace{2mm} 200 \hspace{2mm} g \hspace{2mm} 200 \hspace{2mm} a \hspace{2mm} 300 \hspace{2mm} c' \hspace{2mm} 200 \hspace{2mm} d'\), without \(e\) or \(b\flat\)Translator.]
[22]There is a Chinese tune of the same kind in Ambrosch, loc. cit. vol. i. p. 34, second piece. Another, with a single occurrence of the Sixth, My Peggie is a young thing, may be seen in Songs of Scotland, vol. iii. p. 10. [Scale, \(e \hspace{2mm} 200 \hspace{2mm} f\sharp \hspace{2mm} 300 \hspace{2mm} a \hspace{2mm} 200 \hspace{2mm} b \hspace{2mm} 300 \hspace{2mm} d' \hspace{2mm} 200 \hspace{2mm} e'\), without \(g\) or \(c\). On the bagpipe, see App. XX. sect. K. Probably the scale of the bagpipe has been unaltered since its importation from the East, and it probably never could have played such a scale as it is here supposed capable of performing. — Translator.]
[23] Ambrosch, loc. cit. vol. i. p. 30. To the same class belongs the first piece on p. 35 after Barrow and Amiot. [Scale, \(f \hspace{2mm}200\hspace{2mm} g\hspace{2mm} 200\hspace{2mm} a\hspace{2mm} 300\hspace{2mm} c'\hspace{2mm} 200\hspace{2mm} d'\hspace{2mm} 300\hspace{2mm} f' \).without \(b\flat\), or \(e\). — Translator.]
[24][Taking \(f\sharp\) as the Tonic, the scale would be No. 5, without Second and Fifth thus: $$ f\sharp \hspace{2mm} 300 \hspace{2mm} a \hspace{2mm} 200 \hspace{2mm} b \hspace{2mm} 300 \hspace{2mm} d \hspace{2mm} 200 \hspace{2mm} e \hspace{2mm} 200 \hspace{2mm} f\sharp $$ but taking \(b\) as the tonic the scale would be No. 2 without Second and Sixth, as $$ b \hspace{2mm} 300 \hspace{2mm} d \hspace{2mm} 200 \hspace{2mm} e \hspace{2mm} 200 \hspace{2mm} f\sharp \hspace{2mm} 300 \hspace{2mm} a \hspace{2mm} 200 \hspace{2mm} b $$ which is altogether different. Any reference to tonic, dominant and subdominant, implies harmonic scales, which pentatonic scales could not have been originally. Mr. C. Brown gives this air (Thistle, p. 198) as here printed, but says it varies between his modes of the 3rd (Greek Doric, Ecclesiastical Phrygian) and 5th of the scale (Gr. Ionic, Eccl. Mixolydian). The spelling of the words has been corrected by his edition. — Translator.]

[Adopting the notation explained later on in this chapter, these tetrachords may be accurately written as follows: Nos. 1, 3, 4 and 7 may be played as they stand on the Harmonical, and Nos. 2, 6, 8 by transposition as shewn below, but No. 5 requires the six notes forming 5 perfect Fifths, and these do not occur on the Harmonical, but can be played sufficiently well on any tempered harmonium. Between the names of the notes are inserted the number of cents in the interval between them. By referring to the table called the Duodēnārium, App. XX. sect. E. art. 18, which employs the same notation, the exact position of the notes may be seen, and the correctness of the transpositions verified.

1. Olympos \(b_1 \space 112 \space c' \space 386 \space e_1'\)
2. Old Chromatic \(b_1 \space 112 \space c' \space 70 c_2'\sharp \space 316 \space e_1'\)
          (play \(g \space 112 \space a^1\flat \space 70 \space a_1 \space 316 \space c'\))
3. Diatonic \(b_1 \space 112 \space c' \space 204 \space d' \space 182 \space e_1'\)
4. Didymus \(b_1 \space 112 \space c' \space 182 d_1' \space 204 \space e_1'\)
5. Doric \(b \space 90 \space c' \space 204 \space d' \space 204 \space e'\)
          (not playable on the Harmonical)
6. Phrygian \(d \space 182 \space e_1 \space 134 \space f^1 \space 182 \space g\)
          (play \(g \space 182 \space a_1 \space 134 \space b^1\flat \space 182 \space c'\))
7. Lydian \(c \space 182 \space d_1 \space 204 \space e_1 \space 112 \space f\)
8. Unused \(b_1 \space 112 \space c' \space 274 \space d_2'\sharp \space 112 \space e_1'\)
          (play \(g \space 112 \space a_1\flat \space 274 \space b_1 \space 112 \space c'\))

If the minor Thirds \(d \space f\) and \( e \space g\) were taken as Pythagorean = 294 cents, tetrachord 6 would become \(d \space 204 \space e \space 90 \space f \space 204 \space g\) , which is more intelligible.

On referring to App. XX. sect. D. the ratios corresponding to each of these numbers of cents will be found. — Translator.]

[26][That is, strictly, having a ratio not expressible by whole numbers. — Translator.]
[27][The notes which would form tetrachord 9 might be written in the Translator’s notation, descending from left to right, $$ ^7b\flat \; 85\; a_1 \; 182 \; g \; 231 \; ^7f $$ The three first notes could be played on the Harmonical. The interval 231 cents could be played on it downwards as \(c' \;231 \; ^7b\flat\), but the whole tetrachord cannot be played on it. Here 85 cents represent 21 : 20, while the Pythagorean Semitone 256 : 243 is 90 cents. The difference is small but perceptible. — Translator.]
[28][Using the notation \(^{11}f\) for the 11th harmonic of \(c\), so that \(^{11}\) is equivalent to 33 : 32 or 53 cents, tetrachord 10 may be written downwards: $$ g \; 151 \; ^{11}f \; 165 \; e_1 \; 182 \; d. $$ This is simply, in order, the 12, 11, 10, and 9th harmonic of \(c\), and can be played on the horn or trumpet, and on the 5th octave of the Harmonical, as \(d'''\);\(e_1'''\); \(f'''\); \(g'''\), downwards as \(g'''\); \(f''' \); \(e_1'''\); \(d'''\). The division of the minor Third \(g :e_1 = 316\) cents into 151 and 165 cents is of course only approximative. But it is a purely natural tetrachord of which \(g \; 204 \; f \; 112 \; e_1 \; 182 \; d\) is a deformation. — Translator.]
[29]Journal of the American Oriental Society, vol. i. p. 173, 1847.
[30][If the Octave is divided equally into 24 quarters, each of which is half an equal Semitone or 50 cents, we can write it by using the additional sign \(\flat\)
[31][It is not to be supposed that these two Quartertones, differing only by two cents (32 : 31 = 55 cents, 31 : 30 = 57 cents), were exactly produced. The lutist or lyrist would tune his Fourth \(c : f\) by ear (tolerably correctly), then a major Third below \(f\) or \(d^1\flat\) also by ear (and probably very incorrectly on account of the great difficulty of tuning a major Third), and then would by feeling divide the remaining interval in halves as well as he could. Using \(c:c\)\(\flat\)
[32][Probably the effect was like that which I heard produced by Rája Rám Pál Singh on his Sitár. Here the tone of the note, played by pressing the string against a fret, was sharpened a quarter of a Tone by sliding the finger along the fret (thus deflecting the string and increasing the tension), and then it was allowed to glide on to the proper note by straightening the string without replucking it. I determined the amount of sharpening by observing the distance of deflection, and then, at leisure, measuring by my forks the number of vibrations for the sharpened and normal note, which gave the interval as 48 cents. The effect was very peculiar, but can of course be easily imitated on the violin. On the classical Indian instrument, the Vina, the frets are very high, sometimes about an inch. Hence by pressing down the string behind the fret, the tension could be greatly increased, and as much as a Semitone could be easily added, so that the scale could be indefinitely altered without changing the frets, which were fixed with wax. On the Arabic Rabāb and the curious Chinese fiddles, which have no frets or finger-board, a note could be instantaneously sharpened in a similar manner by pressing more strongly. — Translator.]
[33][And yet a quarter of a Tone is between 2 and 3 commas, and all the difficulties of tuning in just and tempered intonation arise from intervals of a single comma or less. — Translator.]
[34]Even Bellerman is of this opinion (Tonleiter der Griechen, p. 27). Westphal, in his Fragmenten der Griechischen Rythmiker, p. 209, has collected passages from Greek writers proving the real practical use of these intervals. According to Plutarch (De Musica, pp. 38 and 39), the later Greeks had even a preference for these surviving archaic intervals.
[35][See Mr. Rockstro’s article, 'Modes Ecclesiastical,' vol. ii. p. 340, and Rev. T. Helmore's on 'Gregorian Modes,' vol. i. p. 625, in Grove's Dictionary of Music. What Prof. Helmholtz calls the tonic was termed the final. What was the exact intonation of this music it is perhaps impossible to say. Perhaps we may assume it to have been Pythagorean, as $$ d \;204 \;e \;90 \; f \; 204 \; g \; 204 \; a \; 204 \; b \; 90 \; c \; 204 \; d. $$ Of course, modern musicians play them on the piano and organ in equally tempered intonation, as their ancestors played them in meantone intonation. But either of the latter admit of being harmonised; not so the former, so that there is an essential difference. — Translator.]
[36][By a reference to p. 263, note, it will be seen that this paragraph materially alters the intonation from what would result from a mere beginning of each mode with a different note of the Pythagorean or diatonic scale. I therefore repeat the scales as defined by this paragraph in the notation explained on pp.276 to 277 and note footnote 36, and write between each pair of notes the number of cents in the interval between each pair of notes, which will be found useful in future comparisons. These scales should be traced out on the Duodenarium, App. XX. sect. E. art. 18. They cannot be played on the Harmonical.
1. Lydian, \( c \; 182 \; d_1 \; 204 \; e_1 \; 112 \; f \; 204 \; g \; 182 \; a_1 \; 204 \; b_1 \; 112 \; c \)
2. Phrygian, \( d \; 182 \; e_1 \; 134 \; f^1 \; 182 \; g \; 204 \; a \; 182 \; b_1 \; 134 \; c^1 \; 182 \; d \)
3. Dorian, \( e \; 90 \; f \; 204 \; g \; 204 \; a \; 204 \; b \; 90 \; c \; 204 \; d \; 204 \; e\)
4. Hypolydian, \( f \; 204 \; g \; 182 \; a_1 \; 204 \; b_1 \; 112 \; c \; 182 \; d_1 \; 204 \; e_1 \; 112 \; f \)
5. Hypophrygian (Ionic), \( g \; 204 \; a \; 182 \; b_1 \; 112 \; c \; 204 \; d \; 182 \; e_1 \; 134 \; f^1 \; 182\; g \)
6. Hypodoric (Eolic or Locrian), \( a \; 204 \; b \; 90 \; c \; 204 \; d \; 204 \; e \; 90 \; f \; 204 \; g \; 204 \; a \)
7. Mixolydian, \( b_1 \; 112 \; c \; 182 \; d_1 \; 204 \; e_1 \; 112 \; f \; 204 \; g \; 182 \; a_1 \; 204 \; b_1 \)
[37]R. Westphal, in his Geschichte der alten und mittelalterlichen Musik, Breslau, 1864, which is unfortunately still incomplete, uses the previous citations from Aristotle, to frame an hypothesis on the tonic and final cadence of the above scales. But he applies the remarks of Aristotle only to the Doric, Phrygian, Lydian, Mixolydian and Locrian scales, and not to the Eolic and Ionic, which were also known at that time, although the ground for their exclusion is not apparent. In the first four of these be takes the mesē as tonic and the hypatē as the terminal tone. In those scales distinguished by the prefix Hypo-, the hypatē, was both tonic and terminal; but in those having the prefix Syntono-, the hypatē was both the terminal and the Third of the tonic, and the same was the case perhaps for the Boeotian scale, which is only mentioned once. Hence it follows that the minor scale of \(A\) occurs as Doric with the terminal \(e\), as Hypodoric with the terminal \(a\), as Boeotian with the terminal \(c\). Moreover the Mixolydian would be a minor scale of \(E\), with a minor Second, and a terminal in \(b\); the Locrian a minor scale of \(D\) with a major Sixth, and a terminal in \(a\); the Phrygian, Hypophrygian or Iastic, and the Syntonoiastic, major scales of \(G\), with a minor Seventh, the terminals being \(d\), \(g\), and \(b\) respectively. Finally the Lydian, Hypolydian and Syntonolydian would be major scales of \(F\), with superfluous Fourth, and with the terminals \(c\), \(f\), and \(a\) respectively. But according to Westphal the normed major scale was entirely absent. If the Ionic were interpreted according to the words of Aristotle, it would yield a correct major scale. The tonic \(F\) with \(B\) (instead of \(B\flat\) as its Fourth, has a totally impossible appearance to modern musical feeling.
[38][In India there is a highly developed system with a vast variety of scales. — Translator.]
[39][Continuing to use the notation of p. 268, note, these transposed scales may be written as follows. As the order is different from that in p. 267, the numbers there used are added in ( ). The number of cents in each interval will complete the identification. I give only the Ancient Greek names, and the names proposed by Prof. Helmholtz.
1. Lydian — Mode of the First (Major) (1), \(c \;182\; d_1 \;204 \;e_1\; 112\; f\; 204\; g\; 182\; a_1\; 204\; b_1 \; 112 \;c\)
2. Ionic or Hypophrygian — Mode of the Fourth (5), \(c \;204\; d \;182 \;e_1\; 112\; f\; 204\; g\; 182\; a_1\; 134\; b^1\flat \; 182 \;c\)
3. Phrygian — Mode of the minor Seventh (2), \(c \;182\; d_1 \;134 \;e_1\flat\; 182\; f\; 204\; g\; 182\; a_1\; 134\; b^1\flat \; 182 \;c\)
4. Eolic — Mode of the minor Third (Minor) (6), \(c \;204\; d \;90 \;e\flat\; 204\; f\; 204\; g\; 90\; a\flat\; 204\; b\flat \; 204 \;c\)
5. Doric — Mode of the minor Sixth (3), \(c \;90\; d\flat \;204 \;e\flat\; 204\; f\; 204\; g\; 90\; a\flat\; 204\; b\flat \; 204 \;c\)
6. Mixolydian — Mode of the minor Second (7), \(c \;112\; d^1\flat \;182 \;e\flat\; 204\; f\; 112\; g^1\flat\; 204\; a^1\flat\; 182\; b\flat \; 204 \;c\)
7. Syntonolydian — Mode of the Fifth, not in the former table under this name, but really the Hypolydian (4), \(c \;204\; d \;182 \;e_1\; 204\; f_1\sharp\; 112\; g\; 182\; a_1\; 204\; b_1 \; 112 \;c\)
Refer to the Duodenarium, App. XX. sect. E. art. 18. — Translator.]
[40][If we subtract each of the numbers in the names of the modes here proposed, from 9 (reckoning 1 as 8 its Octave), we obtain the numbers on p. 267, which shew the number of the note in the major scale determined by the signature, on which the special scale begins. Thus as 9 less 7 is 2, the mode of the minor Seventh is that numbered 2 on pp. 267, 268. If we call the major scale, when reduced to a harmonisable form, 1. do, 2. re, 3. mi, 4. fa, 5. so, 6. la, 7. ti, then these transformed modes may be called with the Tonic Sol-faists the do, re, mi, &c., modes respectively. — Translator.]
[41][The qualification minor will therefore be always used in this translation, and has been inserted in the above table. — Translator.]
[42][In App. XX. sect. E. No. 10, I have endeavoured to deduce scales for harmonic use, from a general theory of harmony which determines the precise value of each tone as part of a chord, and I have given precise names for them, there exemplified. This harmonic deduction of scales is quite independent of the historical melodic deduction in the text — Translator.]
[43]Geschichte der Griechischen Musik, Berlin, 1855.
[44][See No. 6, of p. 267, text, assuming Pythagorean intonation. — Translator.]
[45]Singularly enough this species of musical scale has been preserved in the Zillerthal in Tyrol, for the wood-harmonicon. This scale has two rows of bars. One forms a regular diatonic scale with the disjunct tetrachord. The other, which lies deep, has the conjunct tetrachord in its upper part.
[46][This seems to be an error for Hypolydian, No. 4 of p. 267, of which the extreme tones are \(f\) and \(f\). — Translator.]
[47]It is by no means an unimportant fact, for our appreciation of the Greek scale, that a flute was found in the royal tombs at Thebes in Egypt (now in the Florentine Museum, No. 2688), which, according to M. Fétis, who examined it, gave an almost perfect scale of Semitones for about an Octave and a half; namely,
Series of primes, \(a \; b\flat \; b \; c' \; c'\sharp \; d'\)
First upper partial tones, \(a' \; b'\flat \; b' \; c'' \; c''\sharp \; d''\)
Second upperpartial tones, \(e'' \; f'' \; f''\sharp \; g'' \; g''\sharp \; a''\)
Third upper partial tones, \(a'' \; b''\flat \; b'' \; c''' \; c'''\sharp \; d'''\)
Representations of such flutes are found in the very oldest Egyptian monuments. They are very long, the holes are all near the end, and hence the arms must have been greatly stretched, giving the player a characteristic position. The Greeks can scarcely have been ignorant of this scale of Semitones. That it was not introduced into their theory till after the time of Alexander, clearly shews the preference they gave to the diatonic scale. [M. Fétis's deductions must be treated with much caution. — Translator.]
[48]von Winterfeld's Johannes Gabrieli und sein Zeitalter, Berlin, 1834, vol. i. pp. 73 to 108.
[49][The following is not an attempt to restore the Greek originals, which have already been treated, but to form harmonic scales on the same, and those are obtained by another process in App. XX. sect. E. art. 9. — Translator.]
[50][I have found much smaller intervals in Chinese instruments. See App. XX. sect. K. — Translator.]
[51][With the subsequent notation and intervals expressed in cents: $$ \begin{gathered} c \; 386 \; e_1 \; 112 \; f \;204 \; g \;182 \; a_1 \; 316 \; c' \\ c \; 386 \; A^1\flat \; 112 \; G \;204 \; F \;182 \; E^1\flat \; 316 \; C \end{gathered} $$ Translator.]
[52][With the subsequent notation and intervals in cents: $$ \begin{gathered} c \; 386 \; e_1 \; 112 \; f \; 204 \; g \; 182 \; a_1 \; 316 \; c' \\ c \; 316 \; e^1\flat \; 182 \; f \; 204 \; g \; 182 \; a_1 \; 316 \; c' \\ c \; 316 \; e^1\flat \; 182 \; f \; 204 \; g \; 112 \; a^1\flat \; 386 \; c' \\ \end{gathered} $$ Translator.]
[53][These are the same three scales as in the last note, read backwards. — Translator.]
[54][In the complete notation, and with intervals in cents, these scales are:
Related to \(c:\; c \;386 \;e_1\; 112 \; f \; 204 \; g \; 182 \; a_1 \; 316 \; c'\)
Related to \(g:\; c \; 204 \; d \; 112 \; e^1\flat \; 386 \; g \; 386 \; b_1 \; 112 \; c'\)
1) Major Scale: \(c \; 204 \; d \; 182 \; e_1 \; 112 \; f \; 204 \; g \; 182 \; a_1 \; 204 \; b_1 \; 112 \; c\)
This is not quite the Greek Lydian, see p. 268, note footnote 38, No. 1. It is 1 \(C\) of App. XX. sect. E. art. 9, 1.
2) The Ascending Minor Scale: \(c \; 204 \; d \; 112 \; e^1\flat \; 182 \; f \; 204 \; g \; 182 \; a_1 \; 204 \; b_1 \; 112 \; c'\)
This is 1. \(C\) (ibid. III).

Related to \(c:\; c \;386 \;A^1\flat\; 112 \; G \; 204 \; F \; 182 \; E^1\flat \; 316 \; C\)
Related to \(g:\; c \; 182 \; B^1\flat \; 316 \; G \; 386 \; E^1\flat \; 112 \; D \; 204 \; C\)
3) The Desending Minor Scale: \(c \; 182 \; B^1\flat \; 204 \; A^1\flat \; 112 \; G \; 204 \; F \; 182 \; E^1\flat \; 112 \; D \; 204 \; C\)
This is not quite the Greek Eolic, see p. 268, note footnote 38, No. 4. It is 1 \(C\) mi.mi.mi (ibid. VIII).
4) Mode of the minor Seventh: \(c \; 182 \; B^1\flat \; 134 \; A_1 \; 182 \; G \; 204 \; F \; 182 \; E^1\flat \; 112 \; D \; 204 \; C\)
This is different from the Greek Phrygian, p. 268, note footnote 34, No. 3, in the two last intervals. It is 1 \(C\) ma.mi.mi. (ibid. VII).

Related to \(c:\; c \;386 \;e_1\; 112 \; f \; 204 \; g \; 182 \; a_1 \; 316 \; c'\)
Related to \(F: c \; 182 \; d_1 \; 316 \; f \; 386 \; a_1 \; 112 \; b\flat \; 204 \; c'\)
5) Mode of the Fourth: \(c \; 182 \; d_1 \; 204 \; e_1 \; 112 \; f \; 204 \; g \; 182 \; a_1 \; 112 \; a_1 \; b\flat \; 204 \; c'\)
This is not quite Greek Ionic or Hypophrygian, p. 268, note footnote 38, No. 2. It is 5 \(F\) (ibid. I).
6) New form of mode of the minor Seventh: \(c \; 182 \; d_1 \; 134 \; e^1\flat \; 182 \; f \; 204 \; g \; 182 \; a_1 \; 112 \; b\flat \; 204 \; c'\)
This is 5 \(F\) (ibid. V).

Related to \(c:\; c \;386 \;A^1\flat\; 112 \; G \; 204 \; F \; 182 \; E^1\flat \; 316 \; C\)
Related to \(F: c \; 204 \; B\flat \; 112 \; a_1 \; 386 \; F \; 386 \; D^1\flat \; 112 \; C\)
7) Mode of the minor Sixth: \(c \; 204 \; B\flat \; 182 \; A^1\flat \; 112 \; G \; 204 \; F \; 182 \; E^1\flat \; 204 \; D^1\flat \; 112 \; C\)
This is not quite the Greek Doric, p. 268, note footnote 38, No. 5. It is mi.mi.mi. (ibid. VIII). -Translator.]
[55]Thus I cannot agree with Hauptmann, in allowing a Pythagorean \(a\), the Fifth above \(d\), in the ascending minor scale of \(c\). D’Alembert introduces the same tone even in the major scale, by passing from \(g\) to \(b\) through the fundamental bass \(d\). But this would indicate a distinct modulation into \(G\) major, which is not required when the natural relations of the tones to the tonic are preserved. See Hauptmann, Harmonik und Metrik, p. 60.
[56]Die Natur der Harmonik und Metrik, Leipzig, 1853, pp. 26 and following. I cannot but join with C. E. Naumann in expressing my regret that so many delicate musical apperceptions as this work contains, should have been needlessly buried under the abstruse terminology of Hegelian dialectics, and hence have been rendered inaccessible to any large circle of readers.
[57][Prof. Helmholtz uses ( - ) between the letters in all such cases. I have taken the liberty from this place onwards, whenever a line or combination of Thirds occurs to leave ( - ) only in the just minor Thirds of 316 cents, to use ( | ) in the Pythagorean minor Thirds of 294 cents, as Prof. Helmholtz does subsequently, and change ( - ) into ( + ) for the major Third of 386 cents. In the case of Fifths which consist of a major and a minor Third 702 = 386 + 316 cents, the symbol is properly \(\pm\) which I here also take the liberty to use. For other intervals I shall use (...) for ( - ), and generally give the precise interval in cents elsewhere. I trust that this change will be found suggestive as well as convenient, and may therefore not be considered presumptuous. — Translator.]
[58][The comma \(81 : 80\) is just over \(21 \small{\tfrac12}\) cents, for which I use 22 cents, see App. XX sect. A. art. 4, and sect. D. Hence a major tone of 204 cents contains about \(9 \small{\tfrac12}\) commas. — Translator.]
[59]In the 1st [German] edition of this book, as in Hauptmann's, the small letters were supposed to be a comma lower than the capital fetters, and a stroke above or below the letters was only occasionally used for raising or depressing the pitch by two commas. Hence a major triad was written \(C \; \text{\textemdash} \; e \; \text{\textemdash} \; G\) or \(\overline{c} \; \text{\textemdash} \; E \; \text{\textemdash} \; \overline{g}\); a minor triad \(a \; \text{\textemdash} \; C \; \text{\textemdash} \; e\), or \(\underline{A} \; \text{\textemdash} \; c \; \text{\textemdash} \; \underline{E}\), &c. The notation used here [in the 3rd and 4th German and the 1st English editions] and also in the French translation is due to Herr A. v. Oettingen. and is much more readily comprehended. [Herr v. Oettingen's notation of lines above and below, which was at Prof. Helmholtz's request retained in the 1st English edition of this translation, was found extremely inconvenient for the printer, and actually delayed the work three months in passing through the press. I have now for some years employed the very easy substitute here introduced. By referring to the table called the Duodēnārium, in App. XX. sect. E. art. 18, where this new notation is systematically carried out for 117 notes, the whole bearing of it will be better appreciated. Another notation which I had used formerly, and into which I translated Herr v. Oettingen's in the footnotes to the 1st edition of this translation, and employed in Table IV., there corresponding to my present Duodenarium, is consequently abandoned, and is now only mentioned to account for the difference in notation between the two editions of this translation. The spirit of Herr v. Oettingen’s notation is therefore retained, while its use has been rendered typographically convenient. — Translator.]

[This variation of the intercalary tones really amounts to a change of mode, so that the names used in the text become ambiguous. This difficulty is overcome by the trichordal notation proposed in App. XX. sect. E. art. 9.

1) The major mode of \(C\) with \(D\), has the 3 major chords \(F + A_1-C, \;C + E_1-G, \;G + B_1- D,\) and is 1 \(C\) But with \(D_1\) in place of \(D\), it has the 3 minor chords \(D_1-F+ A_1, \;A_1- C + E_1, \;E_1-G +B_1\) (of which the two last belong also to the first form), and is therefore 3 \(A_1\) mi.mi.mi. This is a related, but very different, mode.

2) The mode of the Fourth, as it stands in the first line, is not trichordal, but by using \(D\) and \(B^1\flat\) it has the 3 chords \(F+A_1-C, \; C+E_1-G, \; G - B^1\flat + D\), and is hence 1 \(C\) If we take \(D_1\) and \(B\flat\) it has the 3 chords \(B\flat + D_1-F,\; F+A_1-C,\; C + E_1- G\), and is hence 5 \(F\) With both \(D_1\) and \(B^1\flat\), it is again not trichordal.

3) The mode of the minor Seventh. If we take the upper line as it stands, this is also not trichordal. But if we use \(D\) and \(B^1\flat\) it has the 3 chords \(F+A_1-C,\; C - E^1\flat + G, \; G - B^1\flat + D,\) and is hence 1 \(C\) ma.mi.mi. If we take \(D_1\) and \(B\flat\), the 3 chords are \(B\flat + D_1-F,\; F+A_1 - C,\; C-E^1\flat + G\), and the scale is 5 \(F\) With \(D_1\) and \(B^1\flat\), the scale is again not trichordal.

4) Mode of the minor Third. The first line as it stands is not trichordal. Taking \(D\) and \(B^1\flat\) the 3 chords are \(F-A^1\flat + C,\; C- E^1\flat + G,\; G - B^1\flat + D\), and the scale is 1 \(C\) mi.mi.mi. Taking \(D_1\) and \(B\flat\) the 3 chords are \(B\flat + D_1- F,\; F-A^1\flat+ C,\;C - E^1\flat+G\), and the scale is 5 \(F\) ma.mi.mi. With \(D_1\) and \(B^1\flat\) again the scale is not trichordal.

5) Mode of the minor Sixth. The first line as it stands gives the 3 chords \(B\flat-D^1\flat+F,\; F-A^1\flat + C, \; C-E^1\flat + G,\) and the scale is 5 \(F\) mi.mi.mi. If we use \(B^1\flat\) place of \(B\flat\), the 3 chords are \(D^1\flat + F- A^1\flat, \; A^1\flat + C - E^1\flat, \; E^1\flat+G-B^1\flat\), and the scale is 3 \(A^1\flat\) mi.mi.mi.

The modes formed by taking one intercalary tone or the other are therefore quite distinct, though purposely confused in the nomenclature of the text, apparently as an accommodation to the usual tempered notation. — Translator.]

[61][In this case the intonation becomes altogether different. — Translator.]
[62][The fact that the Greek scale was derived from the tetrachord, or divisions of the Fourth, and not the Fifth, leads me to suppose that the tuning was founded on the Fourth and not the Fifth. On proceeding upwards from \(C\) by Fourths, we get \(C \;F \;B\flat \;E\flat\; A\flat \; D\flat \; G\flat \; C\flat \; F\flat \; B\flat\flat \; E\flat\flat \; A\flat\flat \; D\flat\flat \), and on proceeding downwards we get \(C \;G \;D \;A \;E\). Now these notes after \(G\flat\) in the first series, are precisely those of Abdulqadir, written as \(^{\backprime} B_1 \; ^{\backprime} E_1 \; ^{\backprime} A_1 \; ^{\backprime} D_1 \; G_1 \; ^{\backprime} C_1\) on p. 282, according to the notation explained on p. 281, note footnote 63. Of course the Arabic lute, tuned in Fourths, naturally led to this. It is most convenient for modern habits of thought to consider the series as one of Fifths. But I wish to draw attention to the fact that in all probability it was historically a series of Fourths. — Translator.]

[One of the practical results of the Tonic Sol-fa system of teaching to sing the diatonic major scale as marked on p. 274, No.l, in just intonation (see App. XVIII.), has been the discovery that it is not so easy to learn to strike the proper tone by a knowledge of the interval between two adjacent tones in a melodic passage, as by a knowledge of the mental effect produced by each tone of the scale in relation to the tonic. These mental effects are perhaps not very dearly characterised by the mere names given to them in the Tonic Sol-fa books, but the teacher soon makes his class understand them, and then finds them the most valuable instrument which he possesses for inspiring a feeling for just intonation. On these characters of each tone in the (just) diatonic scale, a system of manual signs has been formed, by which classes are constantly led. Particulars are given in 'The Standard Course of Lessons and Exercises in the Tonic Sol-fa Method of Teaching Music, with additional exercises, by John Curwen, new edition, re-written, A.D. 1872'. But it may be convenient to mention in this place the characters and manual signs there given (ib. p. iv.).

I. First step.

Do, Tonic, 'the STRONG or firm tone,' fist closed, horizontal, thumb down.

So, Fifth, 'the GRAND or bright tone,' the fingers extended and horizontal, hand with little finger below and thumb above, so that the palm of the hand is vertical.

Mi, Major Third, 'the STEADY or calm tone,' fingers extended and horizontal, palm of hand horizontal and undermost.

II. Second step.

Re, Second, 'the ROUSING or hopeful tone,' fingers extended, hand forming half a right angle with ground pointing upwards, palm downwards.

Ti, Seventh, 'the PIERCING or sensitive tone,' only the forefinger extended and pointing up, the other fingers and thumb closed, hand forming half a right angle with ground, back of hand downwards.

III. Third step.

Fa, Fourth, 'the DESOLATE or awe-inspiring tone,' only the forefinger extended and pointingdown, at half a right angle with the ground, the back of hand upwards.

La, major Sixth, 'the SAD or weeping tone,' fingers fully extended, whole hand pointing down with a weak fall, back of hand upwards.

It is thus seen that the order of teaching takes the tonic chord first, then the dominant, and lastly the subdominant. The doubtful Second thus comes early on. 'The teacher first sings the exercise to [the names of] consecutive figures, telling his pupils that he is about to introduce a new tone (that is one not do, mi, or so), and asking them to tell him on which figure it falls. When they have distinguished the new tone, he sings the exercise again — laa-ing it [this is calling each note la] — and asks them to tell him how that tone "makes them feel". Those who can describe the feeling hold up their hands, and the teacher asks one for a description. But others, who are not satisfied with words, may also perceive and feel. The teacher can tell by their eyes whether they have done so. He multiplies examples until all the class have their attention fully awakened to the effect of the new tone. This done he tells his pupils the Sol-fa name and the manual sign for the new tone, and guides them by the signs to Sol-fa the exercise and themselves produce the proper effect. The signs are better in this case than the notation, because with them the teacher can best command the attention of every eye and ear and voice, and at the first introduction of a tone, attention should be acute' (ibid. p. 15). This passage, the result of practice with hundreds of thousands of children, shews that a totally new principle of understanding the relation of the tones in a scale to the tonic has not only been introduced, but worked out on a large scale practically, and, as I myself know, successfully. See Prof. Helmholtz's own impression of the success, as long ago as 1864, in App. XVIII. Since that time great experience has been gained and many methods improved. But the object of introducing this notice here is to shew that proper training (such as the ancient Greeks certainly had) could produce the corresponding feeling for the effect of any tone in any scale anyhow divided, independently of the relation of consonances, and that this consideration may help to explain the persistence of many scales which are harmonically inexplicable. No doubt Pythagorean singers hit the degrees of their scale quite correctly, and no doubt the 'mental effects' of their \(A, \; E, \; B\), were very different from those of the harmonisable \(A_1, \;E_1, \; B_1\). We can partially judge of them by the effects of equal temperament, which melodically cannot differ much, although they certainly differ sensibly, from those of Pythagorean intonation. And it must be remembered that singers actually learn to sing in equal temperament, in which all major Thirds are 14 cents too sharp, and then find just major Thirds intolerably flat! To this I would add the following anecdote quoted from Fétis (Hist. Generale de la Musique, vol. ii. p. 27) by Prof. Land (Gamme Arabe, p. 19 footnote), containing 'a fact,' as he says, 'which could not be believed, if it were not attested by the person whom it concerns. The celebrated organist M. Lemmens, who was born in a village of Gampine [or Kempenland, a district in the Belgian province of Limbourg, 51°15'N. lat. 5°20' E. long.], studied music in early youth upon a clavecin (harpsichord), which had been long dreadfully out of tune, because no tuner existed in the district. Fortunately, an organ-builder was summoned to repair the organ at the abbey of Everbode near that village. By chance he called upon the young musician's father, and heard the boy play on his miserable instrument. Shocked at the multitude of false notes which struck his ear, he immediately determined to tune the clavecin. When he had done so, M. Lemmens experienced the most disagreeable sensations, and it was some time before he could habituate his ear to the correct intervals, having been so long misled by different relations.' Hence, false intervals may seem natural. — Translator.]

[64][On this substitution, which amounts to a temperament with perfect Fifths, and major Thirds too flat by a skhisma, or nearly the eleventh of a comma, and which I therefore call skhismic temperament, see Appendix XX. section A. art. 17. It is convenient to use a grave accent prefixed thus \(^{\backprime}E_1\), to shew flattening by a skhisma, and to read it as skhismic, thus, 'skhismic E one'. The above equations can therefore be made precise by writing \(F\flat = ^{\backprime}E_1 , \; C\flat = ^{\backprime}B_1 , \; G\flat = ^{\backprime}F_1\sharp , \&c. \) — Translator.]

R. G. Kiesewetter, Die Musik der Araber nach Originalquellen dargestellt, mit einem Vorworte von dem Freiherrn von Hammer-Purgstall. Leipzig, 1842, pp. 32, 33. The directions given in an anonymous manuscript of the 666th year of the Hegira, a.d. 1267, in the possession of Prof. Salisbury [of Yale Coll.], are essentially the same. See Journal of the American Oriental Society, vol. i. p. 174. [Since the publication of the 4th German edition of this work in 1877, the whole history of the Arabic scale has been reinvestigated from the original Arabic sources by Herr J. P. N. Land, D.D., Professor of Mental Philosophy at Leyden, an Oriented scholar and a musician, and the results were published first in Dutch as a paper in the Transactions of the Dutch Academy of Sciences, division Literature, 2nd series, vol. ix., and separately under the title of Over de Toonladders der Arabische Mustek (on the Scales of Arabic Music) in 1880, and secondly in French as a paper communicated to the International Congress of Orientalists at Leyden in 1882, and published in vol. ii. of their 'Transactions,' and also separately in 1884 as Recherches sur l’histoire de la Gamme Arabe. This paper supersedes in many respects the work of Kiesewetter and von Hammer-Purgstall, of whom the first was a musician but not an Orientalist, and the second an Orientalist but not a musician. Alfārābī's scale was produced by a succession of Fifths [or rather Fourths, see p. 42, note], but a contury and a half previously Zalzal had introduced a new interval 22 : 27 = 355 cents, which Prof. Land terms a neutral Third. It is actually \(\tfrac{12}{11} \times\tfrac{9}{8}\) or 151 + 204 cents, that is, three quarters of a Tone sharper than a major tone, whereas the major Third is 182 cents or a minor Tone sharper, and the minor Third was only a diatonic Semitone 112 cents sharper. The interval 12 : 11 = 151 cents is the well-known trumpet interval between the sharpened Fourth and Fifth, the 11th and 12th harmonics, as may be heard in the Fifth Octave of the Harmonical \(^{11}f''' : g'''\). This on the Arabic lute was necessarily accompanied by a similar interval on the next string, 498 + 355 = 853 cents. These two notes eventually superseded the old Pythagorean minor Third of 294 cents and the Fourth above it of 792 cents; and seem entirely out of the reach of a succession of Fifths or Fourths. But it was the object of Abdulqadir and others to form a succession of Fifths (or rather Fourths) which would include these two intervals, at least approximately. This they accomplished within less than 30 cents by their 384 and 882 cents. It does not appear to have been Abdulqadir’s object to approximate to the just major Third 386, and just major Sixth 884, but to get by means of Fifths or Fourths certain tones which would pass as Zalzal's. The list in the text (p. 282) gives the seventeen tones thus produced with the intervals that they form with each other, and Prof. Helmholtz's names of the notes, completed by a grave accent. Here I re-arrange them in order of Fifths down or Fourths up, the approximate Thirds being added immediately to the right, and the numbers shewing the interval in cents from \(C\):

$$ \begin{alignedat}{5} & E \; &&408\\ & A \; &&906\\ & D \; &&204\\ & G \; &&702\\ & C \; &&0\\ & F \; &&498\\ & B\flat \; &&996\\ & E\flat \; &&294\\ & A\flat \; &&792\\ & D\flat = ^{\backprime}C_1\sharp \; &&90\\ & G\flat = ^{\backprime}F_1\sharp \; &&588\\ & C\flat = ^{\backprime}B_1 \; &&1086\\ & F\flat = ^{\backprime}E_1 \; &&384\\ & B\flat\flat = ^{\backprime}A_1 \; &&882\\ & E\flat\flat = ^{\backprime}D_1 \; &&180\\ & A\flat\flat = ^{\backprime}G_1 \; &&678\\ & D\flat\flat = ^{\backprime}C_1 \; &&1176 \end{alignedat} $$

Observe that the real major Third was the Pythagorean 408 cents, as the minor Third was the Pythagorean 294 cents. Also that 180 cents was within two cents of the minor tone 182 cents. But those approximations were probably not contemplated.

An English concertina, which has fourteen notes to the Octave, was timed with thirteen consecutive Fifths from \(G\flat\) to \(C\sharp\), so that I was able to try the chords \(A \; D\flat \; E , \; DG\flat A\), that is, \(A ^{\backprime} C_1\sharp E , \; D ^{\backprime} F_1\sharp A\), where the major Thirds are two cents too flat, and compare them with the Pythagorean chords \(AC\sharp E , \; DF\sharp A\). The latter were offensive, the former indistinguishable from just. It seems remarkable, therefore, how with such a collection of notes the Arabs escaped harmonic music. But it will be seen on examining the scales formed from them (see especially p. 284, note footnote 69), that they were perfectly unadapted for harmony, which would have occasioned a perfect revolution in their musical systems.

There was certainly no attempt to divide the scale as Villoteau supposed into seventeen equal parts each of about 70.6 cents. For the possible origin of Villoteau's error see infra, p. 520b to 520d'

This system of Abdulqadir prevailed from the thirteenth to the fifteenth century. The modern division into twenty-four Quartertones is noticed on p. 264 and note footnote 29.

The Arabs, however, had also entirely different scales for other instruments than their classical lute, to which alone the above refers. — Translator.]

[66]J. G. L. Kosegarten, Alii Ispahanensis Liber Cantilenarum, pp. 76-86.
[67]Kiesewetter, Die Musik der Araber nach Originalquellen darg., p. 33.
[68][Not therefore according to the forms on p. 268, note, but on the more recent Pythagorean imitation of those forms. They are respectively the representatives of scales 2, 4, and 6 of that note. — Translator.]
[69][In the German text, Quartengeschlecht, or the mode of the Fourth of \(F\). The tones in the mode of the Fourth of \(F\) are those in the Pythagorean scale of \(B\flat\), or, in order of Fifths, \(E\flat \pm B\flat \pm F \pm C \pm G \pm D \pm A\), and the tones of the mode of the minor Seventh of \(F\) are those in the Pythagorean scale of \(E\flat\), or, in order of Fifths, \(A\flat \pm E\flat \pm B\flat \pm F \pm C \pm G \pm D \). The correction is therefore evident. — Translator.]

[Prof. Land (Gamme Arabe, p. 38, note 3) says 'some of the descriptions of Prof. Helmholtz, borrowed from Kiesewetter, do not quite correspond with the original data.' It will be interesting therefore to give these scales as Prof. Land describes them with his (more exact) French orthography of the Arabic names and in his order. The notation is the Translator's, \(^{\backprime}A_1\) being 24 cents flatter than \(A\).

1. 'Ochaq. Our \(F\) major commencing (as shewn by [ ) with the dominant, \(FGAB\flat [CDEF\). 'This commencement is the inevitable consequence of the progression by conjunct tetrachords which belongs to the lute. 'Ochaq is as it were the type of all these maqāmāt, the others of which differ at one time like the tropes or modes of the Greeks and of the middle ages, by the displacement of both the Semitones at once, and at other times like the Greek genera, by exchanges of intervals without disturbing the scheme of two conjunct tetrachords followed by a tone, with the exception of Nos. 7 and 8, which are more distinct from the model maqāma.'

2. Nawā. 'We may say that the scale is that of \(E\flat\) major, beginning with the Sixth.' \(E\flat FGAB\flat [CDE\flat\).

3. Bousilīk or Abou-silīk. 'The scale of \(D\flat\) major beginning at the Seventh, \(D\flat E\flat FG\flat A\flat B\flat [CD\flat\).' The Pythagorean intonation of the three first scales renders them non-harmonic.

4. Rāst. 'The same as 'Ochaq except that the Third \(A\) and the Seventh \(E\) are depressed by a Pyth. comma, \(FG ^{\backprime}A_1B\flat [CD^{\backprime}E_1F\), which makes them just rather than Pythagorean.' The subdominant \(B\flat DF\) is non-harmonic.

5. 'Irāq. 'Like Rāst, but with the second and the sixth above diminished by a Pyth. comma, which makes the second nearly the minor Second 10 : 9, and with grave supplementary Fifth.' \(F^{\backprime}G_1^{\backprime}A_1B\flat ^{\backprime}C_1[C^{\backprime}D_1^{\backprime}E^1F\). This has the proper subdominant \(B\flat ^{\backprime}D_1F\), but the double Fifth is quite non-harmonic.

6. Içfahān. 'Rāst enriched with a grave supplementary Fifth.' \(F^{\backprime}GA_1B\flat ^{\backprime}C_1[CD^{\backprime}E_1F\). Here both the subdominant \(B\flat DF\) and double Fifth render the scale non-harmonic.

7. Zirafkend. \(C^{\backprime}D_1E\flat F^{\backprime}G_1A\flat^{\backprime}A_1^{\backprime}B_1C\). 'An artificial scale composed of fragments of those of \(E\flat\) (\(e\flat f^{\backprime}g_1a\flat c ^{\backprime}d_1e\flat\), Third and Seventh almost just) and of \(C\) (\(c^{\backprime}d_1f^{\backprime}a_1^{\backprime}b_1c\), Second minor and Sixth nearly just) varied also with Pythagorean \(A\) or \(D\) and \(^{\backprime}B_1\).' Of course entirely non-harmonic.

8. Bouzourk. '\(C\) major with the Second, Third, and Seventh diminished by a Pyth. comma, and with a grave supplementary Fifth.' \(C^{\backprime}D_1^{\backprime}E_1F^{\backprime}G_1GA^{\backprime}B_1C\). Both subdominant and dominant are non-harmonic.

9. Zenkouleh. 'Differs from Rāst only in having the Second minor.' \(F^{\backprime}G_1^{\backprime}A_1B\flat [CD^{\backprime}E_1F\). Subdominant non-harmonic.

10. Rāhawi. '\(F\) minor commencing with the Fifth, but with the Sixth and Seventh each increased by a Limma = 90 cents, and the Second diminished by a Pythagorean comma, very nearly our just ascending scale of \(F\) minor.' \(F^{\backprime}G_1A\flat B\flat [C^{\backprime}D_1^{\backprime}E_1F\). The Pyth. scale of \(F\) minor is \(FGA\flat B\flat [CD\flat E\flat F\). Here \(D\flat\) = 90 cents; 90 + 90 = 180 = 204 - 24 cents = \(^{\backprime}D_1\), 24 cents being the Pyth. comma. Similarly \(E\flat\) = 294 cents; 294 + 90 = 384 = 408 - 24 cents = \(^{\backprime}E_1\). Entirely non-harmonic.

11. Hhosaïni. 'Like Nawā, but with the Third and Seventh diminished by a Pyth. comma.' \(E\flat F^{\backprime}G_1AB\flat [C^{\backprime}D_1E\flat\). Entirely non-harmonic.

12. Hhidjāzi. '\(B\flat\) major, beginning with the Second and with the Third, Sixth, and Seventh diminished and therefore nearly just.' \(B\flat [C^{\backprime}D_1 E\flat F^{\backprime}G_1^{\backprime}A_1B\flat\). This is the only one of these scales which is practically harmonic.

If we restore the proper names of the notes in the series of Fifths or Fourths (as in p.281), calculate the cents between each pair of notes and from the first to each note, and begin with the note indicated, we shall have a better idea of the real nature of these scales, thus:

$$\begin{alignedat}{5} &1. \; \; \textit{'Ochaq.} \quad C \; 204 \; D \; 204 \; E \; 90 \; F \; 204 \; G \; 204 \; A \; 90 \; B\flat \; 204 \; C\\ &2. \; \; \textit{Nawā.} \quad C \; 204 \; D \; 90 \; E\flat \; 204 \; F \; 204 \; G \; 204 \; A \; 90 \; B\flat \; 204 \; C\\ &3. \; \; \textit{'Bousilīk.} \quad C \; 90 \; D\flat \; 204 \; E\flat \; 204 \; F \; 90 \; G\flat \; 204 \; A\flat \; 204 \; B\flat \; 204 \; C\\ &4. \; \; \textit{Rāst.} \quad C \; 204 \; D \; 180 \; F\flat \; 114 \; F \; 204 \; G \; 180 \; B\flat\flat \; 114 \; B\flat \; 204 \; C\\ &5. \; \; \textit{'Irāq.} \quad C \; 180 \; E\flat\flat \; 204 \; F\flat \; 114 \; F \; 180 \; A\flat\flat \; 204 \; B\flat\flat \; 114 \; B\flat \; 180 \; D\flat\flat \; 24 \; C\\ \end{alignedat} $$

This double initial \(D\flat\flat,C\) may be compared to our double second in just major scales, and possibly has to be explained in the same way as a real modulation.

$$ \begin{alignedat}{5} &6. \; \; \textit{Içfahān} \quad C \; 180 \; E\flat\flat \; 204 \; F\flat \; 114 \; F \; 204 \; G \; 180 \; B\flat\flat \; 114 \; B\flat \; 180 \; D\flat\flat \; 24\; C\\ &7. \; \; \textit{Zirafkend} \quad C \; 180 \; E\flat\flat \; 114 \; E\flat \; 204 \; F \; 180 \; A\flat\flat \; 114 \; A\flat \; 90 \; B\flat\flat \; 204 \; C\flat \; 114\; C\\ &8. \; \; \textit{Bouzourk} \quad C \; 180 \; E\flat\flat \; 204 \; F\flat \; 114 \; F \; 180 \; A\flat\flat \; 24 \; G \; 204 \; A \; 180 \; C\flat \; 114\; C\\ &9. \; \; \textit{Zenkouleh} \quad C \; 204 \; D \; 180 \; F\flat \; 114 \; F \; 180 \; A\flat\flat \; 204 \; B\flat\flat \; 114 \; B\flat \; 204\; C\\ &10. \; \; \textit{Rāhawi} \quad C \; 180 \; E\flat\flat \; 204 \; F\flat \; 114 \; F \; 180 \; A\flat\flat \; 114 \; A\flat \; 204 \; B\flat \; 204 \; C\\ &11. \; \; \textit{Hhosaïni} \quad C \; 180 \; E\flat\flat \; 114 \; E\flat \; 204 \; F \; 180 \; A\flat\flat \; 228 \; A \; 90 \; B\flat \; 204 \; C\\ &12. \; \; \textit{Hhidjāzi} \quad C \; 180 \; E\flat\flat \; 114 \; E\flat \; 204 \; F \; 180 \; A\flat\flat \; 204 \; B\flat\flat \; 114 \; B\flat \; 204 \; C\\ \end{alignedat} $$

Of these I have been able to play 1, 2, 3 direct, and 4, 5, 10, 12 by transposition upon my Pythagorean concertina (p. 281). When 12 begins with \(B\flat\), or is played by transposition \(a \; b \; d'\flat \; d' \; e' \; g'\flat \; a'\flat \; a'\), it is indistinguishable from the just scale \(a \;b \;c'_1\sharp \; d'\; e'\; f'_1\sharp \;g'_1\sharp \; a'\). The three chords \(d'g'\flat a', \; ad'\flat e', \; e'g'\flat b\) are perfectly good, and the passage \(d'\flat e'a', \; d'g'\flat a', \; e'a'd''\flat, \; e'a'\flat b', \; d'\flat e'a'\) perfectly good, much better than on the piano. Yet it never occurred to Arabs to play in harmony.

'In face of these historical scales,' observes Prof. Land (ibid. p. 38), 'it is difficult to conceive how Kiesewetter could say that the 17 degrees of the complete scale were not treated like sharps and flats, but that each one had the same importance. On the contrary, the 17 degrees were like our 12 Semitones to the Octave, or, still better, like the 17 intervals of the so-called enharmonic scale, which distinguishes sharps and flats, without dividing the Semitones \(E\) to \(F\), and \(B\) to \(C\). To compose their melodies the Orientals, as we do, selected from them several series of 7 [occasionally 8] tones, very slightly different from our diatonic scales.' But so materially different that any attempt to play harmonies upon them would result in frightful dissonance. — Translator.]

[71][If we suppose the pairs of notes in ( ) to have been confused into one by neglecting the Pythagorean comma, then the series of notes on p. 282 becomes \(C\; D\flat \; (^{\backprime}D_1 \; D) \; E\flat \;(^{\backprime}E_1 \; E) \; F \; G\flat \; (^{\backprime}G_1 \;G ) \; A\flat \; (^{\backprime}A_1 \;A) \; B\flat \; ^{\backprime}B_1 \; (^{\backprime}c_1\; c))\), whence the equally tempered scale \(C \;D\flat \; D \; E\flat \; E \; F \; G\flat \; G \; A\flat \; A \; B\flat \; B \; c\) immediately follows. In Meshāqah’s scale of 24 Quartertones, p. 264, that of 12 Semitones is also implicitly contained. — Translator.]
[72]Aristotle, Problems xix. 33. [The passage has already been cited at full, p. 241, note footnote 14. — Translator.]
[73][This cadence is a union of the ancient Doric, beginning with \(c\), rendered harmonisable as the mode of the minor Sixth (\(c...d^1\flat ...e^1\flat ...f...g...a^1\flat ...b^1\flat ... c'\), p. 278, note), with the modern minor, beginning with \(c\), (\(c...d...e^1\flat ...f...g...a^1\flat ... b_1 ... c',\)), and will be more particularly considered in the next chapter, pp. 306 - 308. The intervals expressed in cents are \(D^1\flat\; 386 \;F \;204\; G \;386 \;B_1\), and \(C \;316\; E^1\flat \; 386\; G \;498\; c\). — Translator.]
[74]This periphrasis seems to me to render correctly the last clause in the following citation: Arist Probl. xix. 3,4. [The whole passage may perhaps be translated thus: 'Why do those who sing the parhypatē break down not less than those who sing the nētē and higher tones, though with a greater disagreement (διάστασις)? Is it because they sing this with the greatest difficulty, even when this is the beginning? Does not difficulty arise from straining [and forcing] the voice? This occasions effort, and things done with effort are most apt to fail. But why do they sing the parhypatē with difficulty, and yet take the hypatē easily, although there is only a diesis (Semitone or Quartertone) between them? Is it because the hypatē is sung with a remission of effort, and at the same time it is easy to go upwards after getting oneself together for the effort (σύστασίς)? For the same reason it is easy to sing what leads up to any note, or the paranētē. For the will requires not only conscious thought (σύννοια) but an inclination (κατάστασις) which is perfectly familiar to the habit of mind (ἦθος).' The passage is very difficult, and there was clearly a connection in the writer’s mind between διάστασις, σύστασίς, κατάστασις which influenced his reasoning, but evaporates in translation. — Translator.]
[75][See App. XX. sect. G. art. 6. — Translator.]
[76]Der evangelische Kirchengesang. Leipzig, 1843, vol. i. introduction.
[77]Das Harmoniesystem in dualer Entwickelung. Dorpat und Leipzig, 1866, p. 113.
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